IN  MEMORIAM 
FLORIAN  CAJOR1 


MxL/L^J 


PLANE    AND    SPHERICAL 


TRIGONOMETRY 


BY 


LEVI   L.   CONANT,   Pii.D. 
«? 

PROFESSOR    OF    MATHEMATICS    IN    THE    WORCESTER 
POLYTECHNIC    INSTITUTE 


NEW  YORK  •:.  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


COPYRIGHT,  1909,  BY 

LEVI  L.   CONANT. 

ENTERED  AT  STATIONERS'  HALL,  LONDON. 


CAJOR1 


PREFACE 

IN  this  work  the  author  has  attempted  to  produce  a  text- 
book which  should  present  in  a  concise  and  yet  thorough 
manner  an  adequate  treatment  of  both  the  theoretical  and  the 
practical  sides  of  elementary  trigonometry.  The  material  here 
presented  has  been  gathered  and  tested  during  the  course  of 
many  years  of  experience  in  the  class  room,  and  the  arrange- 
ment and  method  of  presentation  are  the  result  of  numerous 
experiments  made  for  the  purpose  of  ascertaining  what  could 
be  done  most  effectively  in  the  limited  time  usually  devoted 
to  this  subject. 

The  problems  given  in  connection  with  the  different  cases 
under  the  solution  of  triangles  are  nearly  all  new,  and  are  well 
graded  and  sufficiently  numerous  to  give  the  student  ample 
preparation  for  the  various  problems  that  arise  in  plane  sur- 
veying and  in  elementary  astronomical  and  geodetic  work. 
That  portion  of  the  book  which  treats  of  theoretical  trigo- 
nometry has  been  written  in  the  attempt  to  present  this  aspect 
of  the  subject  in  the  simplest  and  clearest  manner,  and  at  the 
same  time  with  the  design  of  equipping  the  student  for  the 
more  advanced  work  in  pure  and  applied  mathematics  which 
is  pursued  in  the  later  years  of  his  college  course. 

The  best  English,  French,  and  Italian  text-books  have  been 
consulted,  as  well  as  those  published  in  this  country.  For 
assistance  in  the  preparation  of  the  work  thanks  are  due  to  my 
colleague,  Professor  Arthur  D.  Butterfield,  to  Professor  W.  B. 
Fite  of  Cornell  University,  to  Professor  O.  S.  Stetson  of  Syra- 
cuse University,  to  Mr.  C.  G.  Brown,  head  of  the  department 
of  mathematics  in  the  Englewood,  New  Jersey,  High  School, 
and  to  Mr.  J.  A.  Bollard,  instructor  in  mathematics  in  the 
Worcester  Polytechnic  Institute. 

LEVI  L.   CONANT. 

WORCESTER  POLYTECHNIC  INSTITUTE, 
WORCESTER,  MASS. 


918256 


ENGINEER'S  TRANSIT,  WITH  GRADIENTEK 
4 


CONTENTS 
PLANE   TRIGONOMETRY 

CHAPTER  PAGES 

I.     THE  MEASUREMENT  OF  ANGULAR  MAGNITUDE       .        .  7-19 

II.     TRIGONOMETRIC  FUNCTIONS  OF  AN  ACUTE  ANGLE         .  20-30 

III.  VALUES  OF  THE  FUNCTIONS  OF  CERTAIN  USEFUL  ANGLES  31-35 

IV.  THE  RIGHT  TRIANGLE 36-50 

V.     THE    APPLICATION    OF    ALGEBRAIC    SIGNS    TO    TRIGO- 
NOMETRY        .        .        .        .        .        .        .        .        .  51-73 

VI.     TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE   .                 .  74-84 
VII.     GENERAL    EXPRESSION    FOR    ALL    ANGLES    HAVING    A 

GIVEN  TRIGONOMETRIC  FUNCTION       ....  85-91 

VIII.     RELATIONS  BETWEEN  THE   TRIGONOMETRIC   FUNCTIONS 

OF  Two  OR  MORE  ANGLES  ......  92-105 

IX.     FUNCTIONS  OF  MULTIPLE  AND  SUBMULTIPLE  ANGLES    .  106-113 

X.     INVERSE  TRIGONOMETRIC  FUNCTIONS       ....  114-121 

XI.     THE  GENERAL  SOLUTION  OF  TRIGONOMETRIC  EQUATIONS  122-130 

XII.     THE  OBLIQUE  TRIANGLE 131-155 

XIII.  MISCELLANEOUS  PROBLEMS  IN  HEIGHTS  AND  DISTANCES  156-165 

XIV.  FUNCTIONS    OF    VERY    SMALL    ANGLES  —  HYPERBOLIC 

FUNCTIONS  —  TRIGONOMETRIC  ELIMINATION       .        .  166-175 

SPHERICAL   TRIGONOMETRY 

XV.    GENERAL  THEOREMS  AND  FORMULAS       ....  177-193 

XVI.     SOLUTION  OF  SPHERICAL  TRIANGLES  194-213 


GREEK   ALPHABET 

Greek  is  written  with  the  following  twenty-four  letters : 


FORM 

NAME 

LATIN 
EQUIVALENT 

A 

a 

alpha 

a 

B 

13 

beta 

b 

r 

7 

gamma 

g 

A 

8 

delta 

d 

E 

€ 

epsilon 

e 

Z 

r 

zeta 

z 

H 

g 

eta 

e 

@ 

0  ></ 

theta 

th 

I 

t 

iota 

i 

K 

/c 

kappa 

c,  k 

A 

\ 

lambda 

1 

M 

p 

mu 

m 

N 

ZV 

nu 

n 

5 

f 

xi 

X 

O 

0 

o  micron 

6 

n 

7T 

pi 

P 

p 

P 

rho 

r 

T 

CT    ? 

T 

sijma 
tau 

s 

\ 
t 

T 

V 

upsilon 

y 

4> 

<#> 

phi 

Ph 

X 

% 

chi 

ch 

¥ 

i/r 

psi 

ps 

n 

CO 

omega 

o 

PLANE  TRIGONOMETRY 

CHAPTER  I 
THE  MEASUREMENT  OF  ANGULAR  MAGNITUDE 

1.  The  size  and  shape  of  a  plane  triangle  can  be  completely 
determined  when  any  three  of  its  six  parts  are  known,  provided 
at  least  one  of  the  known  parts  is  a  side. 

By  means  of  certain  ratios  called  trigonometric  functions, 
which  will  be  defined  later,  trigonometry  enables  us  to  investi- 
gate and  to  determine  the  unknown  parts  and  the  area  of  a  tri- 
angle when  any  three  of  the  parts  are  known,  provided  at  least 
one  of  the  known  parts  is  a  side.  Hence,  in  its  most  elemen- 
tary sense, 

Trigonometry  is  that  branch  of  mathematics  which  treats  of 
the  solution  of  triangles.  During  the  past  two  centuries  the 
sense  in  which  the  word  "  trigonometry  "  is  used  has  been  greatly 
extended,  and  it  is  now  understood  to  include  the  general  sub- 
ject of  mathematical  investigation  by  means  of  trigonometric 
functions. 

Plane  trigonometry  treats  of  plane  triangles,  and  of  plane 
angles  and  their  functions. 

2.  Angles.     In    its   geometric   sense    the  word    "  angle "   is 
defined  as  the  difference  in  direction  of  two  intersecting  lines. 
In  trigonometry,  however,  this  word  receives  an  extension  of 
meaning,  which  must  be  fully  understood  at  the  outset. 

Suppose  two  straight  lines,  OA  and  OB,  are  drawn  from  the 
point  0  in  such  a  manner  that  they  very  nearly  coincide.  Let 
one  of  the  lines,  OA,  remain  fixed  in  position,  while  the  other, 
OB,  revolves  on  the  point  0  as  a  pivot.  We  are  now  free  to 
revolve  OB,  either  back  into  actual  coincidence  with  OA,  or 

7 


8  PLANE   TRIGONOMETRY 

forward,  so  as  to  enlarge  the  opening  between  the  lines.  At 
any  point  of  the  revolution  the  angle  AOB  may  be  said  to 
have  been  formed,  or  generated,  by  the  revolution  of  the 
line  OB. 


In  plane  geometry  angles  greater  than  180°  are  seldom  em- 
ployed, but  in  trigonometry  the  freest  possible  use  is  made 
of  such  angles.  Trigonometry  even  considers  angles  greater 
than  360°,  meaning  by  an  angle  of  that  magnitude  merely  the 
amount  of  revolution  that  has  been  performed  by  the  moving 
or  generating  line. 

As  an  illustration  of  the  meaning  of  the  word  "  angle  "  used 
in  this  sense,  consider  the  movement  of  one  of  the  hands  of  the 
clock.  Let  the  minute  hand  start  from  the  position  it  occupies 
at  noon.  In  fifteen  minutes  it  will  move  over  or  generate  an 
angle  of  90° ;  in  thirty  minutes  an  angle  of  180° ;  in  forty-five 
minutes  an  angle  of  270°  ;  and  in  one  hour  an  angle  of  360°. 
Continuing,  we  may  say  that  in  two  hours  the  minute  hand 
will  move  over  an  angle  of  720°,  in  three  hours  an  angle  of 
1080°,  in  four  hours  an  angle  of  1440°,  in  n  hours  an  angle  of 
rax 360°,  etc. 

Again,  suppose  a  runner  to  be  competing  in  a  two-mile  race 
on  a  circular  track  a  quarter  of  a  mile  in  length.  If  we  sup- 
pose a  line  to  be  drawn  connecting  the  position  of  the  runner 
with  the  center  of  the  circle  formed  by  the  track,  the  position  of 
the  runner  both  on  the  track  and  in  the  race  can  be  described 
at  any  instant  with  perfect  accuracy  by  giving  the  magnitude 
of  the  angle  through  which  this  line  has  revolved  since  the 
beginning  of  the  race. 

Thus,  when  the  line  has  revolved  through  an  angle,  and  hence 
the  runner  has  traversed  an  arc,  of  180°,  he  has  completed  one 
eighth  of  a  mile  ;  when  he  has  traversed  an  arc  of  360°,  he  has 


THE   MEASUREMENT   OF   ANGULAR   MAGNITUDE  !> 

completed  one  fourth  of  a  mile  ;  and  when  he  has  finished 
the  race,  he  has  run  around  the  track  eight  times.  In  other 
words,  when  he  has  finished  the  race  the  line  that  connects  him 
with  the  center  of  the  track  has  revolved  through  an  angle  of 
8  x  360°,  or  2880°.  During  this  time  the  runner  has  traversed 
an  arc  of  the  same  magnitude,  i.e.  of  2880°. 

It  is  at  once  seen  that  an  idea  is  here  introduced  which  is  an 
extension  of  the  idea  of  the  angle  as  it  is  ordinarily  used  in 
geometry.  This  idea,  which  is  fundamental  in  all  work  in 
trigonometry  involving  angles,  is  the  idea  of  formation  or 
generation  in  connection  with  the  angle.  Evidently  a  defini- 
tion of  this  word  is  required  which  differs  from  that  to  which 
the  student  has  become  accustomed  in  geometry  ;  and  in  the 
extended  sense  here  used,  the  term  "angle  "  may  be  defined  as 
follows  : 

An  angle  is  that  relation  of  two  lines  which  is  measured  by 
the  amount  of  revolution  necessary  to  make  one  coincide  with 
the  other. 

3.  The   point  about  which  the  generating  line  revolves  is 
called  the  origin.      The  generating  line  is  called   the  radius 
vector.     The  line  with  which  the  radius  vector  coincides  when 
in  its  original  position  is  called  the  initial  line ;  and  the  line 
with  which  it  coincides  when  in  its  final  position  is  called  the 
terminal  line. 

4.  Positive  and  negative  angles.      It  is  convenient,  and  often 
necessary,  to  know  not  only  the  size  of  an  angle,  but  also  the 
direction  in  which  the  radius  vector  has  moved  while  generating 
the  angle.     For  this  reason  it  is  customary  to  speak  of  angles 
as  being  either  positive  or  negative. 

If  the  radius  vector  moves  in  a  direction  opposite  to  that  of 
the  hands  of  a  watch  when  the  face  of  the  watch  is  toward  the 
observer,  the  angle  it  generates  is  said  to  be  positive.  The 
motion  of  the  radius  vector  as  it  generates  the  angle  is  then 
said  to  be  counter- clockivise. 

If  the  radius  vector  moves  in  the  same  direction  as  the  hands 
of  a  watch  when  the  face  of  the  watch  is  toward  the  observer, 
the  angle  it  generates  is  said  to  be  negative.  The  motion  of 
the  radius  vector  is  then  called  clockwise. 


10 


PLANE   TRIGONOMETRY 


The  angles  AOB1  and  AOB2  are  positive  angles,  and  the 
angles  AOB3  and  AOB4  are  negative  angles.  The  initial  line 
in  each  case  is  OA,  and  the  terminal  lines  are  OBV  OB^  OBS, 
OB^  respectively.  The  direction  of  rotation  for  each  angle 
is  indicated  by  the  arrowhead. . 


5.  Angles  are  often  described    by  referring  them  to  some 
position  with  reference  to  two  intersecting  lines,  at  right  angles 
to  each  other,  of  which  one  is  horizontal  and  the  other  vertical. 
It  is  customary  to  regard  the  horizontal  line  extending  toward 
the  right  as  the  initial  line  for  all  angles,  when  nothing  is  said 
to  the  contrary. 

If  the  radius  vector,  as  shown  in  the  figure,  occupies  any  po- 
sition between  OX  and  OY,  then  the  angle  XOBl  is  said  to  be 

in  the  first  quadrant.  If  the 
radius  vector  is  between  OY 
and  OX',  the  angle  XOB2  is 
said  to  be  in  the  second  quad- 
rant. Similarly,  XOB3  is  said 
to  be  in  the  third  quadrant,  and 
XOB±  in  the  fourth  quadrant. 
These  expressions  only  mean, 
of  course,  that  the  terminal  lines 
lie  in  the  first,  second,  third,  and 
fourth  quadrants  respectively. 

6.  In  practical  work  the  unit  of  measure  that  is  always  em- 
ployed in  dealing  with  angular  magnitudes  is  the  right  angle 
or  some  fraction  of  the  right  angle.     This  unit  is  chosen  because: 

(i)    The  right  angle  is  a  constant  angle. 
(ii)    It  is  easy  to  draw  or  to  construct  in  a  practical  manner, 
(iii)    It   is   the  most   familiar  of  all  angles,   entering  as  it   does  most 
frequently  into  the  practical  uses  of  life. 


THE  MEASUREMENT  OF  ANGULAR  MAGNITUDE    11 

In  geometry  the  right  angle  is  the  unit  universally  used. 
In  trigonometry  two  systems  of  measurement,  involving  the 
use  of  two  different  units,  are  in  common  use. 

7.  The  sexagesimal  system.  In  this  system  the  unit  of 
measure  is  the  right  angle.  The  right  angle  is  divided  into  90 
equal  parts,  called  degrees;  each  degree  is  divided  into  60  equal 
parts,  called  minutes;  and  each  minute  is  divided  into  60  equal 
parts,  called  seconds.  The  symbols  1°,  1',  1",  are  employed  to 
denote  one  degree,  one  minute,  and  one  second  respectively. 

60  seconds  (60")  =  one  minute. 
60  minutes  (60')  =  one  degree. 
90  degrees  (90°)  =  one  right  angle. 

This  system  is  almost  universally  employed  where  numerical 
measurements  are  to  be  made.  It  is,  however,  inconvenient 
because  of  the  multipliers,  60  and  90,  which  it  introduces  into 
computations. 

Another  system,  called  the  centesimal  system,  was  proposed 
in  France  a  little  over  a  century  ago.  In  this  system  the 
right  angle  is  divided  into  100  equal  parts  called  grades,  the 
grade  is  divided  into  100  equal  parts  called  minutes,  and 
the  minute  is  divided  into  100  equal  parts  called  seconds. 
The  centesimal  system  has  been  used  to  some  extent  in  France, 
but  its  use  has  never  been  looked  upon  with  favor  in  other 
countries.  If  its  use  were  to  become  general,  an  enormous 
amount  of  labor  would  have  to  be  expended  in  the  re-computa- 
tion of  existing  tables.  For  this  reason  the  centesimal  system, 
in  spite  of  its  intrinsic  advantage  over  the  sexagesimal  system, 
will  probably  never  come  into  general  use. 


EXERCISE   I 

Express  the  following  angles  in  terms  of  a  right  angle : 

1.  30°.  3.    68°  14'.  5.    228°  46'. 

2.  120°.  4.    114°  38'  12".  6.    321°  14'  22". 


7.  The  angles  of  a  right  triangle  are  in  arithmetical  progres- 
sion, and  the  greatest  angle  is  three  times  the  least ;  what 
is  the  number  of  degrees  in  each  angle  ? 


12 


PLANE    TRIGONOMETRY 


Show  by  a  figure  the  position  of  the  revolving  line  when  it 
has  generated  each  of  the  following  angles  : 

8.  |  rt.  angle.  11.    2^  rt.  angles.  14.     -150°. 

9.  - 1  rt.  angle.  12.    4|  rt.  angles.  15.    275°. 
10.      -1|  rt.  angles.          13.    17|  rt.  angles.            16.    1225°. 

17.  The  angles  of  a  triangle  are  such  that  the  first  contains 
a  certain  number  of  degrees,  the  second  10  times  as  many  min- 
utes,  and  the   third   120   times  as  many   seconds  ;    find  each 
angle. 

18.  How  many  degrees  are  passed  over  by  each  of  the  hands 
of  a  watch  in  one  hour  ? 

Represent  by  a  figure  each  of  the  following  angular  magni- 
tudes : 

19.  l|-+2^  rt.  angles.  23.    4  rt.  angles. 

20.  2|  —  1^  rt.  angles.  24.    4n  rt.  angles  (n  integral). 

21.  -  4  rt.  angles.  25.    (4  n  -f  1)  rt.  angles. 

22.  —  6^  rt.  angles.  26.    (4  n  —  2)  rt.  angles. 

8.  Circular  measure.  Another  system  for  the  measurement 
of  angles  has,  in  modern  times,  come  into  vogue.  It  is  exten- 
sively used  in  work  connected  with 
higher  branches  of  mathematics,  and 
is  the  almost  universal  unit  employed 
in  theoretical  investigations. 

The  unit  of  circular  measure  is  the 
radian,  which  is  obtained  as  follows : 

On  the  circumference  of  a  circle  lay 
off  an  arc,  AB,  equal  in  length  to  the 
radius    of   the  circle,   OA.     The  angle 
AOB  is  called  a  radian.     Accordingly: 

A  radian  is  an  angle  at  the  center  of  a  Circle,  subtended  by  an 
arc  equal  in  length  to  the  radius  of  the  circle. 

In  order  to  use  the  radian  as  a  unit  of  measure,  it  is  necessary 
to  prove  that  it  is  a  constant  angle  ;  or,  in  other  words,  it  is 
necessary  to  prove  that  the  magnitude  of  the  radian  is  the  same 
for  all  circles. 


THE  MEASUREMENT  OF  ANGULAR  MAGNITUDE    13 

9.    THEOREM.      The  radian  is  a  constant  angle. 

By  definition  the  radian  is  measured  by  an  arc  equal  in  length 
to  the  radius.  Also, 

An  angle  of  two  right  angles  is  measured  by  an  arc  equal  to 
one  half  the  circumference. 

Therefore,  since  angles  at  the  center  of  a  circle  are  to  each 
other  as  the  arcs  by  which  they  are  subtended  (Geom.), 

a  radian  radius  R       1 


= =  — 


2  rt.  angles      semi-circumference      irR      TT 
.-.   a  radian  =  ~  of  2  right  angles  =  1  x  180°  =  57.2958° 

=  57°  17' 44. 8"  nearly. 
Therefore  the  radian  is  a  constant  angle. 

10.  The  reason  for  the  use  of  this  unit  may  now  be  readily 
understood. 

Since  1  radian  =  ?_£L^, 

7T 

.•.  TT  radians  =  2  rt,  A  =  180°. 
Similarly,  -  radians  =  1  rt.  Z  =  90°. 

-  radians  =  |  rt.  Z  =  30°. 
6 

£  radians  =  60°. 
o 

1  TT  radians  =  120°. 
f  TT  radians  =  270°. 

2  TT  radians  =  4  rt.  A  =  360°. 
5  TT  radians  =  10  rt.  A  =  900°. 

18  TT  radians  =  36  rt.  A  =  3240°. 

This  gives  a  method  for  the  expression  of  the  value  of  an 
angle  that  is  often  far  more  convenient  than  that  furnished 
by  the  sexagesimal  system.  It  is  especially  useful  in  dealing 
with  angles  of  great  magnitude,  and  it  greatly  simplifies  many 
of  the  investigations  and  formulas  of  trigonometry. 


14  FLAKE   TRIGONOMETRY 

11.  The  symbol  r  is   often   used   as  the  symbol   to   denote 
radians.     Thus,  6r  would  stand  for  6  radians,  Or  for  6  radians, 
7rr  for  TT  radians,  etc. 

When  the  value  of  the  angle  is  expressed  in  terms  of  TT,  and 
when  the  unit  is  the  radian,  it  is  customary  to  omit  the  r  and  to 
give  the  value  of  the  angle  in  terms  of  TT  alone,  the  r  being 
understood.  Thus,  when  referring  to  angular  magnitude,  TT 

means  TT  radians,  ~  means  —  radians,  6  TT  means  6  TT  radians, 

—  2 

etc.  When  the  word  "radians"  is  omitted,  the  student  should 
mentally  supply  it,  or  he  may  readily  fall  into  the  error  of  sup- 
posing that  TT  alone  means  180°.  The  value  of  TT  is  the  same 
here  as  in  geometry,  i.e.  3.14159.  Neither  TT  nor  any  multiple 
of  TT  can  by  itself  ever  denote  an  angle.  It  simply  tells  how 
many  radians  the  angle  contains.  Too  great  care  cannot  be 
exercised  in  keeping  this  distinction  clear. 

12.  To  find  the  number  of  degrees  in  an  angle  containing  a 
given  number  of  radians,  and  vice  versa. 

Since  180°  =  IT  radians, 

1°  =  —  of  a  radian, 
180 

180     ,      , 
and  lr  = ol  a  decree. 


Hence, 

To  convert  radians  into  degrees,  multiply  the  number  of  radians 

i    180 
%—  ' 

To  convert  degrees  into  radians,  multiply  the  number  of  degrees 

by  -E-.  ' 

y  180 

EXERCISE   II 
1.    How  many  degrees  are  there  in  3  radians  ? 


=  3  x         =        =  m.89  nearly 

7T  7T 

=  171°  53'  24"  nearly. 


THE   MEASUREMENT   OF   ANGULAR   MAGNITUDE         15 

2.    How  many  radians  are  there  in  113°  15'  ? 
113°  15'  =  113.25°. 

Since  1°  =  ^, 

180 

113.25°  =  11 3.25  x  -^ 
180 

_  113.25  x  3.14159 

180 

=  1.976  +  radians. 

Express   in   degrees,    minutes,    and   seconds    the   following 
angles: 

5.    ^-  7.    ^-  9.    3?rr. 


Express  in  radians  the  following  angles : 

11.  45°.  14.    225°.  17.    286°  38'.       20.    A°. 

Q0° 

12.  120°.  15.    60°  30'.         18.    684°  26'.       21.    — . 

7T 

13.  135°.  16.    115°  45'.       19.    n°.  22.    78.126°. 

23.  The  difference  between  two  acute  angles  of  a  right  tri- 
angle is  ^  radians;  find  the  value  of  each  of  the  angles  in  degrees. 

5 

24.  If  one  of  the  angles  of  a  triangle  is  56°  and  a  second 

angle  is  ^-^  radians,  find  the  value  of  the  third  angle. 
5  , 

25.  The  angles  of  a  triangle  are  in  A.  P.,  and  the  smallest 
is  an  angle  of  36° ;  find  the  value  of  each  in  radians. 

26.  The  value  of  the  angles  of  a  triangle  are  in  A.  P.,  and 
the  number  of  degrees  in  the  least  is  to  the  number  of  radians 
in  the  greatest  as  60 :  TT  ;  find  each  angle  in  degrees. 

27.  The  value  of  one  of  the  interior  angles  of  one  regular 
polygon  is  to  the  value  of  one  of  the  interior  angles  of  another 
regular  polygon  as  3 :  4,  and  the  number  of  sides  in  the  first  is 
to  the  number  of  sides  in  the  second  as  2  .  3 ;  find  the  number  of 
sides  in  each. 


16 


PLANE   TRIGONOMETRY 


28.  Find  the  number  of  radians  in  one  of  the  interior  angles 
of  a  regular  pentagon  ;  a  regular  heptagon  ;  a  regular  nonagon. 

29.  The  angles  of  a  triangle  are  in  A.  P.,  and  the  number 
of  radians  in  the  least  angle  is  to  the  number  of  degrees  in 
the   mean  angle   as   1:120;   find  the   value   of  each  angle  in 
radians. 

30.  The  angles   of   a   quadrilateral   are  in    A.  P.,  and  the 
greatest  is  double  the  least;  find  the  value  of  each  angle  in 
radians. 

31.  Express  in  degrees  and  in  radians  the  angle  between  the 
hour  hand  and  the  minute  hand  of  a  clock  at  (1)  five  o'clock ; 
(2)  quarter-past  nine  ;  (3)  half -past  ten. 

32.  At  what  time  between  four  and  five  o'clock  are  the  hour 
and  the  minute  hands  of  a  clock  90°  apart  ?     At  what  time  are 
they  180°  apart  ? 

13.    THEOREM.      The  circular  measure  of  an  angle  whose  vertex 
is  at  the  center  of  a  circle  is  the  ratio  of  its  intercepted  arc  to  the 

radius  of  the  circle. 

By  geometry, 

arc  AC 


Z.AOB      arc  A B     a  radius' 

arc  AC '  X/_AOB 
radius 

arc  A  0 

— : —  x  a  radian. 
radius 

Hence,  the  number  of  radians  in  any  angle  is  found  by  dividing 
the  arc  which  subtends  that  angle  by  the  radius  of  the  circle. 

The  formula  just  obtained  is  often  expressed  in  the  following 
convenient,  though  somewhat  incorrect,  form  : 

arc  =  angle  x  radius.  (1) 

The  meaning  of  this  formula  is,  that  the  length  of  any  arc  of  a 
circle  is  equal  to  the  length  of  the  radius  of  the  circle  multiplied 
by  the  number  of  radians  in  the  angle  subtended  by  the  arc. 


THE  MEASUREMENT   OF   ANGULAR  MAGNITUDE         17 


EXERCISE  III 

1.  Find  in  degrees  the  angle  subtended  at  the  center  of  a 
circle  whose  radius  is  30  ft.  by  an  arc  whose  length  is  46  ft.  6  in. 

In  this  circle  the  arc  which  subtends  an  angle  equal  to  a  radian  is  30  ft. 
in  length,  and  the  required  angle  is  subtended  by  an  arc  whose  length  is 
46.5  ft. 

,.  ^radians  =  *??  x  i*°  =  88.8'.     Ans. 
30  300        TT 

2.  In  a  circle  whose  radius  is  8  ft.,  what  is  the  length  of 
the  arc  subtended  by  an  angle  at  the  center,  of  26°  38'  ? 


Let  x  =  the  length  of  the  required  arc. 

ber  of  radians  in  26 
.     (See  Art.  13.) 


Then,  -  =  the  number  of  radians  in  26°  38' 

8 


x  =  3.72  ft.  nearly. 

3.  In  running  at  a  uniform  speed  on  a  circular  track,  a  man 
traverses  in  one  minute  an  arc  which  subtends  at  the  center  of 
the  track  an  angle  of  3|  radians.  If  each  lap  is  880  yd.,  how 
long  does  it  take  him  to  run  a  mile  ? 

Let        x  —  the  number  of  yards  traversed  during  each  minute. 
Then,  x  =  3^  x  R.     (See  Art.  13.) 

99 

=  —  x  140  =  440  yards. 


Since  Y&0  =  4, 

therefore  he  can  run  a  mile  in  4  min. 

4.  The  radius  of  a  circle  is  15  ft.  ;  find  the  number  of  radians 
in  an  angle  at  the  center  subtended  by  an  arc  of  26^  ft. 

5.  The   radius   of   a   circle  is   32  ft.;    find  the  number  of 
degrees  in  a  central  angle  subtended  by  an  arc  of  5  TT  ft. 

6.  The    fly  wheel    of   an   engine    makes    3   revolutions  per 
second  ;  how  long  will  it  take  it  to  turn  through  5  radians  ? 

7.  The  minute  hand  of  a  tower  clock  is  2  ft.  4  in.  long  ; 
through  how  many  inches  does  its  extremity  move  in  half  an 
hour  ? 

CONANT'S  TRIG.  —  2 


18  PLANE   TKIGONOMETRY 

8.  A  horse  is  picketed  to  a  stake ;  how  long  must  the  rope 
be  to  enable  the  horse  to  graze  over  an  arc  of  104.72  yd.,  the 
angular  measurement  of  this  distance  being  150°  ? 

9.  What  is  the  difference  between  the  latitude  of  two  places, 
one  of  which  is  150  mi.  north  of  the  other,  the  radius  of  the 
earth  being  reckoned  as  4000  mi.  ? 

10.  The  angle  subtended  by  the  sun's  diameter  at  the  eye  of 
an  observer  is  32' ;  find  approximately  the  diameter  of  the  sun, 
if  its  distance  from  the  observer  is  92,500,000  mi. 

NOTE.  In  this  example  the  diameter  of  the  sun,  which  is  really  the  chord 
of  an  arc  of  which  the  observer's  eye  is  the  center,  may  be  regarded  as 
coinciding  with  the  arc  which  it  subtends. 

11.  Calling  the  earth  a  sphere,  and  the  arc  of  a  great  circle 
on  its  surface  subtended  by  an  angle  of  1°  at  the  center  69  J  mi., 
what  is  the  radius  of  the  earth  ? 

12.  A  railway  train  is  traveling  at  the  rate  of  60  mi.  an 
hour  on  a  circular  arc  of  two  thirds  of  a  mile  radius ;  through 
what  angle  does  it  turn  in  10  sec.  ? 

13.  The  radius  of  a  circle  is  3  m.;   find  approximately,  in 
radians,  the  arc  subtended  by  a  chord  whose  length 'is  also  3  m. 

14.  How  many  seconds  are  there  in  an  angle  at  the  center 
of  a  circle  subtended  by  an  arc  one  mile  in  length,  the  radius 
of  the  circle  being  4000  mi.  ? 

15.  In  the  circle  of  Ex.  14,  what  is  the  length  of  an  arc  that 
subtends  an  angle  of  3'  at  the  center? 

16.  What  is  the  ratio  of  the  radii  of  two  circles,  if  the  semi- 
circumference  of  the  greater  is  equal  in  length  to  an  arc  of  the 
smaller  which  subtends  an  angle  of  225°  at  the  center? 

17.  If  an  arc  1.309  m.  long  subtends  at  the  center  of  a 
circle  whose  radius  is  10  m.  an  angle  of  7°  30',  what  is  the  ratio 
of  the  circumference  of  a  circle  to  its  diameter  ? 

18.  The  circumference  of  a  circle  is  divided  into  four  parts 
which  are  in  A.  P.,  and  the  greatest  part  is  twice  the  least ;  find 
the  number  of  radians  in  the  central  angle  subtended  by  each 
of  the  respective  arcs  into  which  the  circumference  is  divided. 


THE   MEASUREMENT   OF   ANGULAR   MAGNITUDE         19 

19.  The  diameter  of  a  circle   is  80  m.,  and  an  arc  whose 
length  is  15.75  m.  subtends  a  central  angle  of  22°  30';  find  the 
value  of  TT  to  four  decimal  places. 

20.  How  many  radians  are  there  in  a  central  angle  subtended 
by  an  arc  of  20"  ? 

21.  The  semicircumference  of  a  certain  circle  is  equal  to  its 
diameter  plus  a  given  arc ;  find  the  central  angle  subtended  by 
that  arc. 

22.  Find  the  radius  of  a  globe  such  that  the  distance  of  3  in. 
on  its  surface,  measured  on  an  arc  of  a  great  circle,  may  subtend 
at  the  center  an  angle  of  1°  45'. 

23.  At  what  distance  does  a  telegraph  pole,  24  ft.  high,  sub- 
tend an  angle  of  10',  the  eye  of  the  observer  being  on  the  same 
level  as  the  foot  of  the  pole  ? 

NOTE.  The  suggestion  made  in  connection  with  Ex.  10  applies  to  this 
problem  also.  When  a  chord  and  its  arc  differ  but  little  from  each  other 
it  is  often  convenient  to  use  the  arc  in  place  of  the  chord. 

24.  At  what  distance  will  a  church  spire  100  ft.  high  subtend 
an  angle  of  9',  the  angle  being  measured  from  the  level  on 
which  the  church  stands? 

25.  The  difference  between  two  angles  is  -    -  radians,  and 

y 

their  sum  is  76° ;  what  is  the  value  of  each  of  the  angles  ? 

26.  If  an  incline  rises  5  ft.  in  300  ft.,  find  the  angle  it  makes 
with  its  projection  on  the  horizontal  plane. 

27.  How  many  radians  are  there  in  an  angle  of  a°? 

28.  How  many  radians  are  there  in  an  angle  of  10"  ? 


CHAPTER   II 
TRIGONOMETRIC    FUNCTIONS   OF   AN   ACUTE   ANGLE 

14.  In  the  present  chapter  only  acute  angles  will  be  con- 
sidered. In  Chapter  V  the  definitions  here  given  will  be 
extended  to  angles  of  any  magnitude. 

Let  any  line  having  a  given  initial  position  OA  begin  to 
revolve  on  0  as  a  pivot,  in  a  direction  opposite  to  the  direction 

A'       in  which  the  hands  of  a 
clock  move.    Let  the  angle 
which  it  generates  be  the 
acute  angle  A  OA' . 
A         From     any     point      in 


either  side   of  the   angle, 

asP  in  the  side  OA',  let  fall  a  perpendicular  PM  to  the  other 
side  of  the  angle. 

The  trigonometric  functions,  or  ratios,  of  the  angle  AOA 
are  then  denned  as  follows  : 


The  sine  of  the  angle  A  OA  is  the  ratio      ^  =  side  opposite 

OP        hypotenuse 


The  cosine  of  the  angle  AOA'  is  the  ratio  M  =  side  adjacent 

OP        hypotenuse 


The  tangent  of  the  angle  A  OA  is  the  ratio          =       e  opposite 

OM      side  adjacent 


The  cotangent  of  the  angled  OA  is  the  ratio  °^  =  side  adjacent  § 

MP      side  opposite 

The  secant  of  the  angle  AOAf  is  the  ratio  —  =  hypotenuse  > 

OM.      side  adjacent 


The  cosecant  of  the  angle  A  OA  is  the  ratio  QL  =  h.Ypotenuse 

MP      side  opposite 
20 


TRIGONOMETRIC   FUNCTIONS   OF   AN   ACUTE   ANGLE      21 

In  addition  to  these  there  are  two  other  functions,  less 
frequently  used, 

versed  sine  of  A  OA'  =  1  —  cosine  of  AOA', 
coversed  sine  of  A  OA1  =  1  —  sine  of  A  OA'. 

In  writing,  it  is  customary  to  abbreviate  the  words  "  sine," 
"cosine,"  "tangent,"  etc.,  and  to  express  the  functions  of  any 
given  angle,  a?,  as  follows : 

sin  x,    cos  x,    tan  a?,    cot  a?,    sec  #,    esc  #,    vers  a?,   covers  x. 

It  should  be  noted  at  the  very  beginning  that  these  functions 
are  mere  numbers,  and  their  values  can  be  expressed  numerically 
whenever  the  angle  to  which  they  belong  is  known.  Thus, 
sin  x  may  equal  J,  J,  or  any  other  proper  fraction ;  tan  x 
may  equal  2,  5,  18,  or  any  other  real  number  whatever.  The 
expression  sin  a;,  for  example,  is  a  single  symbol,  and  is  to 
be  regarded  as  the  name  of  the  number  which  expresses  the 
value  of  the  particular  ratio  in  question.  The  expressions 
sin,  cos,  etc.,  have  no  meaning  unless  some  angle  is  asso- 
ciated with  them. 

15.  The  trigonometric  functions  are  always  constant  for  the 
same  angle. 

From  any  points  in  either  side  of  the  angle  x,  as  A, 
Af,  A",  drop  perpendiculars  AB,  A'B',  A"B"  to  the  other 
side.  Then,  by  geometry,  the  triangles  A  OB,  A' OB1,  A"  OB" 
are  similar,  and  their  homologous  sides  are  proportional. 
Therefore,  A 

BA_B'A'  _B"A"  _ 
OA~  OA'=''  OA"  = 

OB      OB'      OB" 


OA      OA'      OA" 


=  cos  a:, 


U  B"  A       B 

and  similarly  for  the  other  functions. 

Hence,  the  value  of  any  function  of  x  remains  unchanged  as 
long  as  the  value  of  the  angle  itself  remains  unchanged. 

Any  increase  or  decrease  in  the  size  of  the  angle  pro- 
duces a  change  in  the  value  of  the  function,  or  ratio.  From 
this  it  is  readily  seen  why  these  ratios  are  called  functions  of 
the  angle. 


22  PLANE   TRIGONOMETRY 

From  the  -last  paragraph  the  following  important  results  may 
now  be  stated : 

(1)  To  every  acute  angle  there  corresponds  one  and  only  one 
value  of  each  trigonometric  function. 

(2)  Two    unequal   acute    angles    have    different    trigonometric 
functions. 

(3)  To  each  value  of  any  trigonometric  function  there  is  but 
one  corresponding  acute  angle. 

16.  Fundamental  relations  between  the  trigonometric  func- 
tions of  an  acute  angle.  From  the  definitions  given  in  Art.  14 
it  follows  immediately  that  the  sine  of  the  angle  x  is  the  recip- 
rocal of  the  cosecant  x ;  also  that  cosine  x  is  the  reciprocal  of 
secant  #,  and  that  tangent  x  is  the  reciprocal  of  cotangent  x. 
That  is,  . 

cscrr' 

cos#  = ,    or  cos  #  sec  #  =  1, 

sec  # 

tan  x  = ,  or  tan  ^  cot  a:  =  1. 

cot  x 

Also,  it  follows  from  the  definitions  that 

sin  x          T  cos  a: 

tan#=-   — ,   and          cot  x  —  — 

cosx  sin  a; 

In  the  right  triangle  ABC,  a2  +  b2=  c2.     Therefore, 


c  , 

and  _  =  !  +  — 

a2  a2 


From  these  equations  it  follows  that  ^ 

sin2  a?  +  cos2  x  -  1  ,  (3) 

sec2  x  —  1  +  tan2  x,  (4) 

esc-'  JT,  =  1  +  cot2  x.  (5) 


TRIGONOMETRIC  FUNCTIONS  OF  AN  ACUTE  ANGLE  23 

17.  From  the  definitions  of  the  trigonometric  functions, 
p.  20,  it  follows  that  in  any  right  triangle  any  function  of 
either  of  the  acute  angles  is  equal  to  the  corresponding  co- 
function  of  the  other -acute  angle.  For  example, 

sinJ.  =  -,  and  cos£=-.     .-.  sin  A  —  cos  B=  cos(90°  —  A}. 
G  c 

Similarly,  cos  A  =  sin  B  =  sin  (90°  —  J.), 

tan  A  =  cot  B  =  cot  (90°  -  J.), 
cot  A  =  tan  B  =  tan  (90°  -  A), 
sec  A  =  esc  B  =  esc  (90°  -  A), 
esc  A  =  sec  B  =  sec  (90°  -  A), 
vers  A  =  covers  B  =  covers  (90°  —  A), 

covers  A  =  vers  B  =  vers  (90°  —  A). 
Hence, 

Any  function  of  an  acute  angle  is  equal  to  the  corresponding  co- 
function  of  its  complement. 

The  meaning  of  the  prefix  co,  in  cosine,  cotangent,  cosecant, 
and  coversed  sine  appears  from  the  above.  The  cosine  of  an 
angle  is  the  complement-sine,  i.e.  the  sine  of  the  complement  of 
that  angle:  the  tangent  of  an  angle  is  the  cotangent  of  its  com- 
plementary angle;  and  a  similar  statement  may  be  made  for 
the  secant  and  for  the  versed  sine  of  an  angle. 

ORAL  EXERCISES 
Prove  the  following  relations  : 

1.  sin  A  cot  A  =  cos  A. 

SOLUTION.     Using  only  the  left  number  of  the  equation,  we  proceed  as 

follows :  ___  A 

sin  A  cot  A  =  sin  A  -.  (Art.  16,  (2).) 

sin  A. 

=  cos  A . 

.*.  sin  A  cot  A  =  cos  A . 

2.  cos  A  tan  A  =  sin  A. 

3.  (sec  A  —  tan  A) (sec  A  +  tan  A)  =  1. 

4.  (esc  A  —  cot  ^4)(csc  A  -f-  cot  A)  =  1. 

5.  (tan  A  +  cot  A)  sin  A  cos  ^4  =  1. 


24  PLANE   TRIGONOMETRY 

6.  (tan  A  —  cot  A)  sin  A  cos  A  =  sin2  A  —  cos2  A. 

7.  sin2  0  -*-  esc2  6  =  sin4  0. 

8.  sin40-  cos40=sin20-cos20. 

9.  (sin  0  -  cos  0)2  =  1  -  2  sin  0  cos  0. 

10.  (sin  0  -  cos  0)2  +  (sin  0  +  cos  0)2  =  2. 

11.  sec  0  cot  0  =  esc  0. 

12.  (tan  0  +  cot  0)2  =  sec2  0  +  esc2  (9. 

13.  cot2  9  cos2  0  =  cot2  0  -  co.,2  0. 

14.  sin2  0  +  esc2  6  +  2  =  (sin  (9  +  esc  0)2. 

15.  vers  6  (1  +  cos  0)  =  sin2  0. 

16.  sin2  (9  +  vers2  (9  =  2(1-  cos  (9). 

17.  sec  0  —  sin  0  tan  0  =  cos  0. 

18.  esc  6  —  cos  0  cot  6  =  sin  0. 

19.  sec2  (9  -  tan2  0  =  sin2  0  +  cos2  0. 

20.  esc2  (9  -cot2  0  =  sin2  (9  +  cos2  0. 

EXERCISE  IV 
Prove  the  following  identities  : 

1.  cos4  d  -  sin4  0  =  2  cos2  6-1. 
SOLUTION.     Using  only  the  left  member  of  the  equation,  we  proceed  as 


=  (1)  (cos2  B  -  sin2  0) 

=  cos2  0  -  (1  -  cos2  0)     (Art.  16,  (3).) 

=  2cos20-  1. 


2.    sin3  0  +  cos3  0  =  (sin  0  +  cos  0)(1  -  sin  0  cos  0). 

3         smA     +1+008^2080^. 
1  +  cos  A         sin  A 

4.  (1  +  sin  a  +  cos  a)2  =  2(1  +  sin  a)(l  +  cos  a). 

5.  (COS3  0  -  sin3  0)  =  (cos  0  -  sin  0)(1  +  sin  0  cos  0). 

6.  cos2  /3  (sec2  /3-  2  sin2  /3)  =  cos4  /3  +  sin4  /3. 


.< 


t 

TRIGONOMETRIC   FUNCTIONS   OF   AN   ACUTE   ANGLE      25 

sin  (S         1  4-  sin  /3  9  ,., 

7.  :: :~ 5  +  H  =  sec2ff  (csc/3  +  1). 

1  —  sin  /8         sin  £ 

8.  tan  a  +  tan  /3  =  tan  a  tan  /:?  (cot  a  +  cot  /3). 

9.  cot  a  4-  tan  /3  =  cot  a  tan  /3  (tan  a  +  cot  /3). 
10.    cos6  a  4-  sin6  a  =  1  —  3  sin2  a  cos2  a. 


/I  —  sin  A 

11.  \/7 —        —  =  sec  A  —  tan  A. 

*1  4-  sin  A 

12.  sin2<9  tan204-cob20  cot2  0  =  tan2  0  4- cot2  <9- 1. 

13.  CSC/     +     CSC/     =2  sec2  A. 
esc  JL  —  1      esc  J.  -h  1 


esc  A 

14.  • —  —  =  cos  ^L. 
cot  A  +  tan  A 

15.  (1  —  sin  a  —  cos  «)2(1  +  sin  a  +  cos  a)2  =  4  sin2  a  cos2  a. 

16.  (sec  A  H-  cos  A)  (sec  A  —  cos  J.)  =  tan2  A  +  sin2  ^. 

17. — =  sin  A  cos  A. 

cot  ^4.  H-  tan  A 

1  —  tan  A  _  cot  J.  —  1 
1  +  tan  A      ccff-A+1' 

19.  sjn3  J.  cos  J.  +  cos3  A  sin  ^4  =  sin  A  cos  A.       * 

20.  sin2  J.  cos2  ^L  +  cos4  A  =  1  —  sin2  A. 
esc  a  —  sec  a       cot  a  —  tan  a 


cot  a  -f-  tail  a       esc  a  +  sec  a 
1  +  tan2  A      sin2  A 


21. 


22. 


23.  sec  A-tauA      l_2sQGA  tan  A  +  2  tan2  ^ 
sec  J.  4-  tan  ^4. 

24.  tan2  a  sec2  a  4-  cot2  a  esc2  a 

=  sec4  a  esc4  a  —  3  sec2  a  esc2  a. 

tan  .A  cot  A 

25. T  +  ^ —  =  sec  ^4.  esc  y±  4- 1. 

1  —  cot  A      1  —  tan  ^1 

cos  A  sin  ^4.  ,,    ,  ,, 

26.  -—  —  =  sin  A  +  cos  A. 
1  —  tan  A      1  —  cot  A 


26  PLANE   TRIGONOMETRY 

27.      COt4  A  +  COt2  A  =  CSC4  A  —  CSC2  A. 


28.  Vcsc2  A  —  I  =  cos  A  esc  ^4. 

29.  tan2  ^4  -  sin2  A  =  sin4  .A  sec2  A. 

30.  (1  +  cot^4.—  csc  A)  (I  +  tan  J.  +  sec  A)  =2. 

1  11  1 


31. 


32. 


csc  A  —  cot  A      sin  .A      sin  A      csc  .A  +  cot  A 
cot  .A  cos  A.        cot  A  —  cos  vl 


cot  A  +  cos  A       cot  A  cos  ^4. 

33.  2  -  vers2  0  =  sin2  0+2  cos  0. 

34.  sin8  A  —  cos8  .A  =  (sin2  A  —  cos2  A)(l  —  2  sin2  A  cos2  A). 
cos  ^1  esc  A  -  sin  A  sec  ^ 


3g 


cos  ^   +  sin  ^. 

tan  A  +  sec  ^4.  —  1  _  1  +  sin  A 

tan  A  —  sec  A  +  1         c<  >s  A 


37.  (tan  a  +  csc  £)2  —  (cot  fi  —  sec  a)2 

=  2  tan  a  cot  /3(csc  a.+  sec  /3) 

38.  2  sec2  a  —  sec4  a  —  2  csc2  a  +  csc4  a  =  cot4  a  —  tan4Tx. 

39.  (sin  a  +  csc  a)2  +  (cos  a  +  sec  a)2  =  tan2  a  +  cot2  a  -f  7. 

v  x  .  sec  A        csc,  A 

40.  (1  +  cot  A  +  tan  A)  (sin  J.  —  cos  A)=  - 

csc2  J.      sec2  A 


2 


41.    2  vers^l  -f  cos2  J.  =  1  +  vers 


42.  S6C  x  ~  tan  *  =  1  +  2  tan  x  (tan  g  -  sec  x). 
sec  a;  +  tan  x 

2  sin  6  cos  0  —  cos  6 

43.  -  -  —  -  -  r  =  COL  (7. 


44.  (sin  a  cos  /3  +  cosa  sin  /3)24-  (cos  a  cos  /S  —  sin  a  sin/^)2=l 

45.  (ta,^  +  sec^  =  l±^. 

1  —  sin  6 


TRIGONOMETRIC   FUNCTIONS  OF   AN   ACUTE   ANGLE      27 

18.  Limits  of  the  values  of  the  trigonometric  functions  of  an 
acute  angle. 

Since  sin2  A  +  cos2  A  =  1, 

and  since  each  term,  being  a  square,  is  positive,  neither  sin2  A 
nor  cos2  A  can  be  greater  than  unity.  Hence,  neither  sin  A  nor 
cos^l  can  be  numerically  greater  than  unity. 

Since  esc  A  is  the  reciprocal  of  sin  A,  and  sec  A  is  the  recip- 
rocal of  cos  A,  both  sec  A  and  esc  A  can  have  any  values  numeri- 
cally greater  than  unity,  but  neither  can  ever  be  numerically 
less  than  unity. 

Since  sec2  A=\  +  tan2  J., 


tan  A  =  Vsec2^.—  1. 

Hence,  tan  A  can  have  any  value  between  0  and  oo.  And 
since  cot  A  is  the  reciprocal  of  tan  A,  therefore  cot  A  can  have 
any  value  between  oo  and  0. 

These  results  are  summarized  as  follows  : 

When  A  is  an  acute  angle, 

sin^L  can  take  any  value  between  0  and  +  1, 
cos  A  can  take  any  value  between  -f-  1  and  0, 
tan  A  can  take  any  value  between  0  and  +00, 
cot  A  can  take  any  value  between  +  GO  and  0, 
sec  A  can  take  any  value  between  -f  1  and  -foo, 
esc  A  can  take  any  value  between  +00  and  +  1. 

These  results  also  follow  directly  from  the  definitions  of  the 
functions  of  an  acute  angle,  p.  20. 

19.    To  express  all  the  trigonometric  functions  in 
terms  of  any  one  of  them. 

From  any  point  in  either  side  of  the  angle 
A  let  fall  a  perpendicular  upon  the  otlu  r 
side.    Let  the  hypotenuse  of  the  right 
triangle  thus  formed  be  taken  as  unity, 


28  PLANE   TRIGONOMETRY 

and  call  the  perpendicular  a.     Then  the  remaining  side  of  the 
right  triangle  is  Vl  —  a2.     Then, 

sin  A  =  -  —  a  =  sin  A, 


cos  A  =  Vl  —  a2  =  Vl  —  sin2  A, 
sin  A 


tan  A  = 


Vl  -  a2      Vl  -  sin2  A 


cot  A  = 


Vl  -  sin2  A 
sin  A 


sec  .   = 


Vl  -  a2      Vl  -  sin2 


csc  .    =  -= 


,  — 
sin  .A 


This  gives  the  value  of  each  of  the  functions,  except  the  vers  A 
and  the  covers  A,  in  terms  of  sin  A. 

To  express  the  values  of  the  functions  in  terms  of  cos  J., 
tan  A,  or  of  any  other  given  function  of  A,  proceed  in  a  similar 
manner.  It  is  not  best,  however,  to  assume  the  hypotenuse 
equal  to  unity  for  all  cases.  Sometimes  the  side  opposite  the 
given  angle  should  be  taken  as  unity,  and  sometimes  the  side 
c  adjacent.  For  example,  to  find  the 
other  functions  of  A  in  terms  of 
tan  A,  assume  the  side  adjacent  A 
equal  to  unity,  and  let  the  side  oppo- 
A  B  site  the  same  angle  equal  a ;  then  the 

hypotenuse  of  the  right  triangle  equals  VI  +  a2,  and  the* required 
values  are  found  as  follows  : 

tan  A  =  -  =  a  =  tan  A, 


Vl  +  a2      Vl  +  taii2^. 


-V 


TRIGONOMETRIC   FUNCTIONS   OF   AN   ACUTE   ANGLE     29 
cos  .4  = - = , 


tan 


a  tan  A 

In  this  work  it  will  be  noticed  that  the  side  adjacent  to  A  is 
assumed  equal  to  unity,  while  in  the  preceding  the  hypotenuse 
was  assumed  to  be  unity.  Any  other  supp6sition  might  be 
made  with  equal  correctness,  but  no  other  would  be  equally 
convenient. 

EXERCISE  V 

1.    Express  all  the  other  functions  of  6  in  terms  of  cos  6. 

This  problem  can  be  solved,  and  the  required  values  found, 
in  a  manner  similar  to  that  employed  in  finding  the  values  of 
each  of  the  other  functions  in  terms  of  sin  0,  or  tan  0,  which 
has  just  been  illustrated.  Or  the  values  can  be  found  by  means 
of  the  relations  deduced  in  Art.  16.  Thus  : 


=  vi-  cos2  6, 
si  ii#      Vl  —  cos 


tan  0  = 

cos  0  cos  0 

»„„ A  «~~  £ 

:,etc. 

Sill 


2.  Express  all  the  other  functions  of  0  in  terms  of  cot  6. 

3.  Express  all  the  other  functions  of  0  in  terms  of  sec  0. 

4.  Express  all  the  other  functions  of  6  in  terms  of  esc  6. 

5.  Given  sin  6  =  f ,  find  cos  6  and  tan  0. 


cos 


B  =  Vl  -  sin2  0  =  Vl  -  ^  =  i  V21. 


cos0     5      5  5      V21      V21      21 


30 


PLANE   TRIGONOMETRY 


6.  Construct  the  angle  6  if  tan  6  =  f  . 

The  angle  6  may  be  considered  one  of  the  acute  angles  of  a  right  triangle. 
Hence,  to  construct  6  we  have  only  to  construct  a  right  triangle  whose  legs 
are  respectively  2  and  7.  Since  tan  0  =  f  ,  0  is  the  acute  angle  opposite  the 

Bide  2'  11. 

7.  If  sin   0  =  -j3_,  find  sec  0. 

8.  If  sin  A  =  J-J,  find  vers  A. 

9.  If  cos  0  =  f,  find  esc  6  and  tan  0. 

10.  If  cos  0  =  |,  find  cot  0  and  sec  0. 

11.  If  tan  A  =  J-J,  find  sec  J.  and  cos  A. 

12.  If  tan.  A  =  |,  find  esc  A  and  cos  .4. 

13.  If  cot  A  =  |  ,  find  sin  J.  and  cos  A. 

14.  If  sec  B  =  5,  find  siri  J5  and  tan  .#. 

15.  If  sec  B  =  |^,  find  tan  .5  and  vers  B. 

16.  If  esc  A  =  8,  find  cos  A  and  tan  A. 

17.  If  esc  JL  =  f  ,  find  sin  A  and  sec  A. 

18.  Find  all  the  functions  of  each  of  the  acute  angles,  ^4,  B, 
of  the  right  triangle  whose  sides  are  8,  15,  17. 

19.  Find  all  the  functions  of  each  of  the  acute  angles,  A,  B, 
of  the  right  triangle  whose  sides  are  x  +  y, 


2  xu 


x  —  y     x-y 

20.  .  If  sin2  0  +  cos  0  =  2|,  find  tan  0. 

21.  If  tan2  0  -  sec  6  =  f>,  find  cos  0. 

22.  If  10  sin2  6  -  5  cos  0  =  -  |,  find  esc  6. 

23.  If  sin  0  + cot  0  =  4y,  find  0080. 


24.    If  sin  0,  =  a  and  tan  6  =  ft,  prove  (1  -  a2)(l  -  ft2)  =  1  . 


' 


\  \ 


CHAPTER   III 

VALUES   OF   THE   FUNCTIONS   OF   CERTAIN   USEFUL 

ANGLES 

20.   Functions  of   an  angle  of  0°.     If  the  angle   A   is  very 
small,  then  in  considering  the  value  of  sin  A,  that  is,  the  ratio 

CB 

-  —  ,  it  is  at  once  seen  chat  the  numerator,  CB,  is  very  small  in 


comparison  with  the  denominator,  AB.     Hence,  the  numerical 
value  of  sin  A  is  very  small  when  the  angle  A  is  very  small. 
Also,  if  A  decreases,  the  numerator  of  the  fraction  will  also 
decrease,    while    the    denominator    will 
remain    constant  ;    and    as    the    angle 
approaches  0°  as  a  limit,  the  sine  of  the 

angle  will  also  approach  0  as  a  limit.  When  the  angle  becomes 
0°,  that  is,  when  A  B  coincides  with  A  C,  we  shall  have  CB  =  0, 
and  AB  =  AC.  Hence, 

sin  0°=  -=0, 


When  we  say  that  sin  A  —  0  when  A  =  0°,  we  simply  mean 
that,  if  A  is  made  small  enough,  we  can  make  the  value  of  CB, 
and  hence  the  value  of  sin  A  as  small  as  we  please  ;  or,  to  ex- 
press the  same  statement  in  different  words,  we  can  make  sin  A 
smaller  than  any  assignable  quantity. 

Hence,  as  stated  above,  sin  A  approaches  0  as  a  limit  when 
A  approaches  0°  as  a  limit. 

In  a  similar  manner,  we  interpret  the  statements  cosO°  =  l, 
tan  90°  =  oo  ,  etc,,  as  meaning  that  cos  A  approaches  1  as  a  limit, 
tan  A  approaches  oo  as  a  limit,  etc.,  when  A  approaches  0°  as  a 
limit,  when  A  approaches  90°  as  a  limit,  etc. 

31 


32 


PLANE   TRIGONOMETRY 


21.  Functions  of  an  angle  of  30°.  Let  OAC  be  an  equilateral 
triangle  ;  then  is  it  also  equiangular.  From  the  vertex  0  draw 
OB  perpendicular  to  AC.  Then  in  the  right  triangle  GAB  the 
angle  A  =  60°,  and  the  angle  A  OB  =  30°.  Also, 


theleg  AB=IAC=± 
Let  AB  =  a.     Then        OA  =  2  a,  and 


=  V  6U2  -  AB*  =  V4  a2  -  a2  =  V3  a2  =  a  V3. 


The  trigonometric   functions  of  30°  can   now  be    found   as 
follows : 


OA 


tan  30°=  BA  •  =  -A_  =  JL  =  1  V3, 
av/3      V3      3 


BA 


V3 


esc  30°  =  —  =  —  =  2. 
BA       a 


22.  Functions  of  an  angle  of  45°.  Let  OAB  be  an  isosceles 
right  triangle.  Each  of  the  acute  angles  is  45°,  and  the  leg  OB 
equals  the  leg  AB.  ^4 

Let  AB  =  a.     Then  OB  =  a  and  OA  =  a V2, 
and  we  have : 


sin  45' 


cos45°=^  = 


- 
2 


FUNCTIONS   OF   CERTAIN    USEFUL   ANGLES 


-BA-a--\ 
~          ~' 


33 


CSC45°=^=±     «  =  V2. 
BA        a 

23.  Functions  of  an  angle  of  60°.  Let  OAC  be  an  equilateral 
triangle.  Then  is  it  also  equiangular.  From  the  vertex  A 
draw  AB  perpendicular  to  00.  Then  in  the  right  triangle 
OAB,  angle  0=  60°,  and  angle  OAB=  30°.  Also,  OB  =  ±OC 
=  \  OA. 

Let    OB  =  a.      Then    OA  =  2  a,    and   AB  =  V  OA*  -  OB* 


=  V4«2_a2=V3a2  =  6?V3. 

The  trigonometric  functions  of  60°  can   now  be  found  as 
follows : 

*„.!£.•£     V5,i     - 


OB        a 


cot60°  =  ^^  =  -^=-JL  =  i 
BA     aV3      V3      3 


fO  4          9  /,  9          9 

csc  60°  -  ^  =  ^^  =-^-  =  £• 


O         a 


24.  Functions  of  an  angle  of  90°.  Let  the  angle  AOB  (p.  34) 
be  very  nearly  a  right  angle.  Then  the  angle  A  is  very  small, 
and  B  the  foot  of  the  perpendicular  from  A  to  OB  is  very  near 
the  vertex  0.  When  the  angle  0  approaches  a  right  angle,  AB 


CON  A  NT'S    TKIO.  -  3 


34 


PLANE   TRIGONOMETRY 


will  approach  coincidence  with  AO,  and  B  will  approach  coinci 
dence  with  0.     Hence, 

.        o      BA      OA     - 


YA 


one 

cos  90   =  —  —  = 


0 


OA      OA 
BA      OA 


0, 


_ 

BA      OA 


*  =-^=ir 

OA     OA 


The  real  meaning  of  these  equations  is  that,  as  the  angle  ap- 
proaches 90°  as  a  limit,  the  sine  of  the  angle  approaches  1  as  a 
limit,  the  cosine  approaches  0  as  a  limit,  the  tangent  approaches 
oo  as  a  limit,  etc.  It  is,  however,  customary  to  say  sin  90°  =  1, 
cos  90°  =  0,  tan  90°  =  oo  ,  etc. 

A  more  complete  discussion  of  the  values  of  the  trigonometric 
functions  for  limiting  cases  such  as  the  above  is  given  later. 
See  Art.  41,  p.  59. 

25.  In  the  following  table  are  collected  the  results  obtained 
in  the  last  five  sections.  These  results  are  exceedingly  im- 
portant, and  the  student  should  become  thoroughly  familiar 
with  them  before  proceeding  further. 


0° 

30° 

45° 

60° 

90° 

sine 

0 

1 

JvS 

|V3 

1 

cosine 

1 

JV3 

*V2 

i 

0 

tangent 

0 

iV3 

1 

V:] 

CO 

cotangent 

CO 

V3 

1 

*V3 

0 

secant 

1   I  jVS 

V2 

o 

-a 

cosecant 

cc    j      2 

V2 

|v3 

1 

FUNCTIONS   OF   CERTAIN    USEFUL   ANGLES  35 

It  is  necessary  to  commit  to  memory  only  one  half  of  this 
table.  The  remainder  can  be  obtained  at  any  time  by  means 
of  the  relations  which  were  found  in  Art.  17,  of  which  the 
-following  is  a  condensed  statement  :  Any  trigonometric  function 
of  an  acute  angle  is  equal  to  the  corresponding  co-function  of  its 
complement. 

EXERCISE  VI 

Verify  the  following  : 

,     1.  cos  0°  +  sin  30°  +  sin  90°  =  2£. 

2.  cos  0°  cos  60°  -f  sin  0°  sin  60°  -f  sin  30°  =  1. 

3.  tan2  30°+  sec2  30°  =  If. 

4.  cos2  60°  +  cos2  45°  4-  cos2  30°  =  f  . 

5.  sin  60°  cos  30°  -f-  cos  60°  sin  30°  =  1. 

6.  sin2  30°  tan2  45°  +  sec2  60°  sin2  90°  =  4  J. 

7.  (sin  30°  +  cos  60°)  (sec  45°  +  c.sc  45°;  -  2  V2. 

8.  sin  30°  sin  45°  sin  60°  tan  60°  =  f  V2. 

9.  cot  30°  tan  60°  sin  45°  cos  45°  =  f  . 

10.  tan2  45°  +  sin2  30°  -  cos2  30°  -  f  tan2  30°  =  J. 

Prove  the  following  identities  : 

11.  sin  A  cos  (90°  -  A)  sec  (90°  -A)  =  sin  A. 

12.  cos  A  cos  (90°  -  A)  sin  (90°  -  A)  esc  A  ==  cos2  A. 

13.  tan  (90°  -  A)  cot  (90°  -  A)  tan  A 

=  cos  (90°-  A)  esc  (90°-  A). 

cos  (90°  -A)    cot(90°-J.) 

14.  —  =  sin  A. 
esc  (90°  -  A)            sin  A 

15.  cos2  A  sec2  (90°  -  A)  tan2  A  cot2  (90°  -A)  =  tan2  A. 

16     tan2  (90°-^)  cos2  .4  csc2(90°-^)  ^       4  A 

sin2  A         '  cot2  (yO°  -  A)  '  sec2  (yo°  -  A)  " 

17.  cos  (90°  -  A)    1  -  cos  (90°  -  A)  _  t.m  A 

covers  A  sin  ((JO  —  A) 

18.  secMEEl  +cos2(90°-^)csc2(90°-7l). 

19.  csc2^  =  1  +  sin2  (90°  -  A)  sec2  (90°  -  A). 

20     cot2  (90°  -  A)    tan2  (90°  -A')^1 
esc2  (90°  -  A)  '  sin2  (90°  -A) 


CHAPTER   IV 


THE   RIGHT   TRIANGLE 

26.  In  order  to  solve  a  right  triangle,  two  parts  besides  the 
right  angle  must  be   given,  of  which  at  least  one  must  be  a 
side.     The  known  parts  may  be  : 

1.  An  acute  angle  and  the  hypotenuse. 

2.  An  acute  angle  and  the  opposite  leg. 

3.  An  acute  angle  and  the  adjacent  leg. 

4.  The  hypotenuse  and  either  leg. 

5.  The  two  legs. 

27.  In  the  preceding  sections  we  have  seen  that  the  trigono- 
metric functions  are  pure  numbers ;  and  in  the  case  of  the  angles 
0°,  30°,  45°,  60°,  and  90°,  the  values  of  these  functions  have  been 
ascertained.      From  a   trigonometric  table  the  values  of   the 
functions  of  any  angle  can  be  found ;  and  by  the  aid  of  these 
values  the  solution  of  any  triangle  can  be  effected. 

The  method  for  each  case  arising  under  right  triangles  is 
illustrated  by  the  following  examples : 


CASE  1 
Given  A  =  61°  22',  c  =  46.2;  find  B,  a,  b. 


fit  ° 22' 


B  =  90°  -  61°  22'  =  28°  38'. 


b 


(1) 

(2)  sin  A---    .-.a=csiuA 

c 

=  46.2  x  0.8777. 
.-.  a  =  40.54. 

(3)  cos^l  =-.    .-.  b  =  coos  ,4 

c 

=  46.2  x  0.4792. 
.-.  b  =  22.14. 

30 


THE   RIGHT   TRIANGLE 

CASE  2 
Given  A  =  31°  17',  a  =  321 ;  find  B,  c,  b. 

(1)  B  =  90°  -  31°  17'  =  58°  43'. 

(2)  sin.4  =  ^    .'•€  =  -;?— 


321 


=  618.14. 


0.5193 
.-.  c  =  618.14. 

(3)  tan  J.  =  -•  /.  b  =  — — 
b       tan  A 

321 


0.6076 
.-.  b  =  528.31. 


CASE  3 
Given  A  =  43°  42',  b  =  38.6  ;  find  B,  a, 


,  43'42' 


(1) 


=  90° -43°  42' =46°  18'. 


(2)  tan  /I  =  -  •    .•.  a  —  b  tan  A 


(3) 


=  38.6  x  0.9556. 
.-.  a  =  36.89. 

* 


cos  A 


_    38.6 
0.7230* 

.-.  c  =  53.39. 


CASE  4 

Given    a  =  36. 4,  <?  =  48.4;  find  A,  B,  b. 
(1)    sin  A  =  - 

=  36.4 
~48.4 

=  0.7521,  nearly. 
...  A  =  48°  46'. 


(2) 


B  =  90°  -  48°  46'  =  41°  14'. 


(3)   tanJ  =-•     .-.6  = 
b 

36.4 


tan 


1.141 
.-.  b  =  31.9. 


38 


PLANE   TRIGONOMETRY 


The  value  of  b  could  also  be  found  directly  by  means  of  the  familiar 
geometric  relation 

from  which  we  have  b  =  Vc2  —  a2. 

CASE  5 
Given   a  =  34.9,  b  =  38.6  ;  find  A,  B,  c. 

(1)  tan  4  =  2 

=  34.9 
38.6 

=  0.9041. 
.-.  A  =42°  7'. 

(2)  £  =  90°  -42°  7' =  47°  53'. 


b-38.6 


(3)    sin,4  =?.     .-.  c  =  -^— 
c  sin  ^1 

34.9 


0.6706 
.-.  c  =  52.04. 


The  value  of  c  can  also  be  found  directly  by  means  of  the  relation 
c2  =  Va*  +  b'2. 

From  the  methods  of  solution  illustrated  in  the  examples 
given  in  Art.  27,  we  deduce  the  following  general  rule : 

Rule  for  the  solution  of  right  triangles.  From  the  equation 
A  +  B  =  90°,  and  from  the  equations  that  define  the  functions 
of  an  acute  angle  of  a  right  triangle,  select  an  equation  in  which 
the  required  part  is  the  only  unknown  quantity.  From  this  equa- 
tion find  an  expression  for  the  required  part,  and  compute  the  value 
of  this  part  from  the  expression  thus  obtained. 

If  a  and  c,  or  b  and  c,  have  values  that  differ  but  little 
from  each  other,  the  methods  here  given  will  yield  inaccurate 
results.  In  such  cases  the  method  of  Art.  101,  p.  144,  should 
be  employed. 

The  student  will  find  it  advantageous  to  check  his  results  in 
all  cases,  to  avoid  numerical  errors  as  far  as  possible.  Any 
method  of  checking  can  be  employed  that  involves  a  process 
of  solution  different  from  the  one  used  in  first  obtaining  the 
required  part. 


THE    RIGHT   TRIANGLE 


39 


EXERCISE  VII 

In  the  following  examples,  use  the  first  two  parts  as  the  given 
parts,  and  solve  for  the  three  remaining  parts  : 

~i.   A  =  21°  19',     c  =  18.        £=68°  41',   a  =6.5, 

=  49°  16', 


2. 

^  =  40°  44', 

£=31. 

3. 

^  =  71°  38', 

£=5.4. 

4. 

£=13°  14', 

£  =  92. 

5. 

.A  =  63°  11', 

a  =  12. 

6. 

B  =  43°  52', 

6  =  70. 

7. 

A  =  19°  36', 

6  =  42. 

8. 

5  =  56°  17', 

a  =  9. 

9. 

a  =  12.6, 

£  =  26. 

10. 

6  =  42.6, 

£  =  46. 

6  =  16.8. 

=  20.2,  «  =  23.5. 

£  =  18°  22',   a  =5.12,  6  =  1.7. 

^4  =72°  46',    6  =  21,  a  =  89.6. 

,6  =  26°  49',    6  =  6.1,  c  =  13.4. 

^  =  46°8',     a  =72.8,  c  =  101. 

B  =70°  24',   a  =15,  £=44.6. 

^4  =  33°  43',    6  =  13.5,  £  =  16.2. 

A  =  28°  59',  B  =  61°  1',  6  =  22.7. 

A  =  22°  10',  B  =  67°  50',  a  =  17.4. 


SOLUTION  BY  LOGARITHMS 

28-  Problems  in  the  solution  of  triangles  can  usually  be  per- 
formed quite  as  expeditiously  by  the  use  of  logarithms  as  by  the 
use  of  the  actual  values  of  the  trigonometric  functions,  and  in 
many  cases  the  amount  of  labor  is  very  greatly  reduced  by  the 
use  of  logarithms. 

The  method  of  solution  by  logarithms  in  the  different  cases 
that  arise  in  connection  with  right  triangles  is  illustrated  by 
the  following  problems : 

CASE  1 

Given  ^1  =  59°  17',   £=42.68;  find  £,  a,  ft, 

(1)  B  =  90°  -  59°  17'  =  30°  43'. 

(2)  sin  A=--      .-.a  =  csiuA.  n 

log  a  =  log  c  +  log  sin  A. 
logc  =  1.63022 
log  sin  A  =  9.93435  -  10 
log  a  =  1.56457 
.-.  a  =  36.69. 

(3)  cos  A  =--,    b  =  ccosA. 

log  c  =  1.63022 

log  cos  A  =  9.70824  -  10 

log  b  =  1.33846 

.-.  b  =  21.8. 


40 


PLANE   TRIGONOMETRY 


Given  A  =  55 


55  °//' 


CASE  2 
a  =  68. 34;   find  B,  b,  c. 

(1)  B  =  90°  -  55°,  11'  =  34°  49'. 

(2)  tanX=2.     ...&=_«_ 

b  tan  yl 

log  ft  =  log  a  -f  colog  tan  A. 
log  «  =  1.83467 
colog  tan  A  =  9.84227  -  10 
log  b  =  1.67694 
.-.  b  =  47.527. 


(3)  sin  A  =  1 


sin  A 

log  c  =  log  «  +  colog  sin  A. 

log  a  =  1.83467 

colog  sin  A  =  0.08567 

logc=  1.92034 

.-.  c  =  83.242. 


CASE  3 
Given  A  =  49°  13',  b  =  72.3  ;  find  B,  a,  e. 

(1)  73  =  90°  -49°  13'  =  40"  17'. 

(2)  tan  A  =-.     .-.  a  =  ft  tanA. 

log  a  =  log  ft  +  log  tan  A. 
log  ft  =    1.85914 
log  tan  .4  =  10.06116  -  10 
logo=    1.92330-10 

a  =  83.81. 


(3)  cos  4  =  - 
c 


cos  4 

log  c  =  log  b  4-  colog  cos  A . 
log  ft  =1.85914. 
colog  cos  A  =  0.18495 
logc  =  2.04409 
.-.  c  =  110.68 

CASE  4 
Given  c=  61.14,  a= 48.56;  find  ,4,^, 

(1)    sin  A  =a-- 
c 
log  sin  A  =  log  a  +  colog  c. 

loga  =  1.68628 
colog  c  =  8.21367  -  10 
log  sin  /I  =  9.89995  -  10 
.-.  A  =  52°  35'. 


*72  3 


THE   RIGHT   TRIANGLE 


41 


(2)     cos  A  —  -. 


.'.  b  =  c  cos  A  . 

log  b  =  log  c  +  log  cos  A, 
logc  =  1.78633. 
log  cos  A  =  9.78362  -  10 
log  b  =  1.56995 
b  =  37.149. 


(3)    tan  B  =  - 


lo 


tan  B  =  log  b  -f  colog  a. 


log  b  =  1.56995 
colog  «  =  8.31372  -  10 
log  tan  B  =  9.88367  -  10 
.-.  B  =  37C  25'. 

CASE  5 
Given   a  =  126,    b  =  198  ;  find  A,  B,  c. 


(1) 


log  tan  ^4  =  log  a  +  colog  5. 

log  a  =  2.10037 
colog  b  =  7.70333  -  10 
log  tan  A  =  9.80370  -  10 
.-.4  =  32°  28'. 


(2)   tanB=- 


log  tan  B  =  log  6  +  colog  a. 

log  6=    2.29667 
colog  a=    7.89963-10 
log  tan  B  =  10.19630  -  10 
.-.  B  =  57°  32'. 


sin  A  =  - 


C  =  ~  -  7' 

sm  -4 

log  c  =  log  a  +  colog  sin  A. 

log  a  =  2.10037 

colog  sin  A  -  0.27018 

log  c  =  2.  37055 

.-.  c  =  234.72. 

NOTE.  In  the  last  two.  cases  the  angle  B  might  have  been  found  directly 
by  subtracting  A  from  90°.  It  is,  however,  better  to  determine  the  value  of 
the  second  angle  independently,  as  a  means  of  checking  the  work. 


42  PLANE   TRIGONOMETRY 

AREA   OF   THE   RIGHT   TRIANGLE 

29.  The  area  of  any  triangle  is  equal  to  one  half  the  product 
of  the  base  and  the  altitude.  In  the  case  of  the  right  triangle 
either  of  the  legs  can  be  regarded  as  the  base  and  the  other  as 
the  altitude.  Hence  the  area  of  a  right  triangle  can  be  found 
when  any  two  parts  are  known,  provided  one  or  both  the  known 
parts  are  sides,  by  computing,  if  necessary,  the  legs  of  the  tri- 
angle, and  then  taking  one  half  their  product.  That  is, 

If  a,  6,  denote  the  legs  of  a  right  triangle,  and  A  the  area, 

then  A  =  i-  ab.  (1) 

Ex.  l.  In  the  right  triangle  ABO,  given  .4  =  36°  14', 
a  =  26. 8;  to  find  the  area. 

First  find  log  b  by  the  method  of  Case  2,  p.  40.     Then  we  have 
log  A  =  log  a  -f  log  b  +  colog  2. 
log  a  =  1.42813 
log  b  =  1.56315 
colog  2  =  9.69897  -  10 
log  A  =  2.69025 
.-.  A  =  490.06. 

Ex.  2.  In  the  right  triangle  ABC,  given  ^.  =  40°  23, 
c  =  39.6;  to  find  the  area. 

First  find  log  a  and  log&  as  in  Case  1,  p.  89.     Then  we  have 
log  A  =  log  a  +  log  b  -f  colog  2. 
log  a  =  1.40921 
log  b=  1.47950 
colog  2  =  9.69897-  10 
log  A  =  2.58768 
.-.  A  =  386.97. 

EXERCISE   VIII 

Solve  the  following  right  triangles,  finding  the  angles  to  the 
nearest  minute : 

1.  Given  A  =  34°  10',     a  =  21 ; 

find      ^  =  55°  50',     5  =  30.939,     c=  37.39. 

2.  Given  5=50°  12',     a  =65; 

find      .4=39°  48',     b  =  78?15,       c=  101.55. 


THE   RIGHT   TRIANGLE  43 

3.  Given  5  =  47°  15',  c=  54.39; 

find      .4  =  42°  45',  a  =  36. 92,      5  =  39.94. 

4.  Given  A  =  31°  25',  c  =  45.62  ; 

find      B  =  58°  35',  b  =  38.93,     a  =  23.78. 

5.  Given  A  =  29°  17',  c=31.68; 

find      £=60°  43',  a  =  15.495,   6=^=27.63. 

6.  Given  4  =  49°  17',  c=  36.48; 

find      JB=40°43',  a  =  27.65,      6=23.796. 

7.  Given  J.  =  41°  9',  b  =  156; 

find      B  =48°  51',  a  =136.33,    c=  207.17. 

8.  Given  B  =  59°  11',  6  =  221 ; 

find      ^1=30°  49',  a  =131. 83,    ^  =  257.33. 

9.  Given  B  =  62°  55',  c=92.4; 

find      ^1  =  27°  5',  a  =  42.068,    6  =  82.268. 

10.  Given  A  =  29°  31',  a  =  290.6; 

find      B=  60°  29',  b  =  513.29,    c=  589.85. 

11.  Given  ^  =  45°  20',  a  =  41. 46; 

find      ^.  =  44°  40',  5  =  41.946,    c  =  58.979. 

12.  Given    a  =20. 08,  c?=28.26; 

find      A  =  45°  17',  ^  =  44°  43',  b  =  19.885. 

13.  Given  B  =  55°  13',  a  =  72.96  ; 

find      ^1  =  34°  47',  6  =  105.04,    c  =  127.89. 

14.  Given  B  =  51°  19',  6  =  106.8; 

find      A  =38°  41',  a  =85.512,   c=  136.81. 

15.  Given  B  =  59°  49',  a  =  254.36 ; 

find      4  =30°  11',  6  =  437.33,    c  =  505.92. 

16.  Given  A  =  51°  50',  6  =  6.813; 

find      B  =  38°  10',  a  =  8.668,      c=  11. 025. 

17.  Given  B  =  57°  46',  6  =  0.0688; 

find      A  =  32°  14',  a  =  0.04338,  c  =  0.08134. 


44  PLANE   TRIGONOMETRY 

18.  Given    6  =  963.3,  c=1465; 

find      ^1  =  48°  53',          £=41°  7',  a  =  1103.7. 

19.  Given   a  =  691,  e=  877.62; 

find      .4  =  51°  56',         j5=38°4',  6  =  541.05. 

20.  Given    a  =  62. 36,  6  =  33.823; 

find      A  =  61°  32',         ^  =  28°  28',         c  =  70.96. 

In  the  following  examples  find  the  required  angles  to  the 
nearest  second  : 

21.  Given  A  =  41°  38'  20",    b  =  262.38  ; 

find      .g  =  4S021' 40",    a=233.27,          <?=351. 08. 

22.  Given  ^=71°  14'  12",     <?=  129.3; 

find      ^=18°  45' 48",    a  =  122.43,          6  =  41.6. 

23.  Given  A  =  41°  17'  30",    a  =  29.41; 

find      B  =  48°  42'  30",    6  =  33.486,          c  =  44.568. 

24.  Given  B  =  61°  12'  15",    c  =  382.6 ; 

find      A  =  28°  47'  45",    a  =  1 84. 29,          6  =  335. 29. 

25.  Given    6  =  1426,  c  =  2291.2; 

find      A  =  51°  30'  38",  B  =  38°  29'  22",  a  =  1793.38. 

26.  Given  B  =  54°  2'  28",      a  =  49.628  ; 

find      A  =  35°  57'  32",    6  =  68.41,  c  =  84.514. 

27.  Given    «  =  35.421,  6  =  18.168; 

find      A  =  62°  50'  40",  B  =  27°  9'  14",    c  =  39.81. 

28.  Given    a  =  39.313,  6  =  19.852; 

find      A  =  63°  12'  26",  J5  =  26°  47'  34",  c  =  44.036. 

29.  Given    a  =  126. 43,  6=131.52; 

find      A  =  43°  52'  9",     B  =  46°  7'  51",     c  =  182.44. 

,    * 

30.  Given    a  =  476.32,  c  =  812.36; 

find      .4  =  35°  53'  53",  B  =  54°  6'  7",      6  =  658.05. 

31.  Given  ,4  =  68°  17'  22",    c  =  269.4; 

find      B  =  21°  42'  38",  a  =  250.29,          6  =  99.658. 


THE    RIGHT   TRIANGLE  45 

111  the  following  ten  examples  find  the  area  of  the  triangle  in 
each  case,  having  given  : 

32.  a  =  10,  6  =  12.  37.  .4  =  42°  27',  6  =  50. 

33.  a  =  268,          b  =  316.  38.  A  =  54°  24',  c  =  90. 

34.  a  =  3,  <?=5.  39.  £=39°  55',  a  =294. 

35.  b  =  20. 7844,  ^=24.  40.  B  =  66°  36',  b  =  48. 

36.  J.=  35°,          a  =  16.  41.    ^=70°  52',  <?  =  582. 

42.  Find  the  value  of  A  in  terms  of  a  and  c. 

43.  Find  the  value  of  A  in  terms  of  a  and  A. 

44.  Find  the  value  of  A  in  terms  of  a  and  B. 

45.  Find  the  value  of  A  in  terms  of  c  and  A. 

46.  Given  A  =  72,  a  =  9  ;  find  A. 

47.  Given  A  =  72,  5  =  9;  find  A. 

48.  Given  A  =  250,         A  =  40° ;         find  a. 

49.  Given  A  =  250,    B  =  29°  30' ;  find  a. 

50.  Given  A  =  254.2,  <?=  32;   find  B. 

51.  The  hypotenuse  of  a  right  triangle  is  28  and  one  of  the 
legs  is  13.     Find  the  angle  opposite  the  given  leg. 

52.  The  legs  of  a  right  triangle  are  36  and  39,  respectively. 
Find  the  angle  opposite  the  shorter  leg. 

53.  The  tangent  of  one  of  the  acute  angles  of  a  right  tri- 
angle is  2\.     Find  the  angle. 

54.  The  cotangent   of   one  of   the  acute  angles  of   a  right 
triangle  is  \\.      What  is  the  angle? 

55.  One  of   the  acute  angles  -of  a  right  triangle  is  49°  38' 
and  the  adjacent  leg  is  68.42.     Find  the  hypotenuse  and  the 
other  leg. 

56.  The  legs  of  a  right  triangle  are  41625.3  and  11362.7, 
respectively.     Find  the  larger  angle. 

57.  The  hypotenuse  of  a  right  triangle  is  262.46  and  one  of 
the  acute  angles  is  28°  15'  42".     Find  the  opposite  leg. 


46  PLANE  TRIGONOMETRY 

58.  The  legs  of  a  right  triangle  are  515.38  and  221.34,  re- 
spectively.    Find  the  hypotenuse. 

59.  One  of  .the  acute  angles  of  a  right  triangle  is  46°  21'  and 
the    adjacent   leg    is    26.38.      Find    by  natural    functions  the 
other  leg  and  the  hypotenuse. 

Angle  of  elevation  and  angle  of  depression.     The    angle   of 
elevation  of   an  object  above  the  point  of  observation  is  the 
D  angle  between  a  line  from  the  eye 

angle  of  depression  \^          of  the  observer  to  the  object  and 

a  horizontal  line  in  the  same  ver- 
tical plane.    The  angle  of  depres- 
sion of  an  object  below  the  point 
angle  of  elevation  of  observation  is   the   angle   be_ 

tween  a  line  from  the  eye  of  the 

observer  to  the  object  and  a  horizontal  line  in  the  same  vertical 
plane. 

In  the  figure,  BA  C  is  the  angle  of  elevation  of  the  point  B 
above  the  point  A ;  and  DBA  is  the  angle  of  depression  of  the 
point  A  below  the  point  B. 

60.  The  angle  of  elevation  of    the    top  of   a   tower  80  ft. 
high  is  41°  49'.     What  is  the  distance  of  the  point  of  observa- 
tion from  the  foot  of  the  tower  ? 

61.  At  a  distance  of  31.15  ft.  from  the  foot  of   a  vertical 
cliff  the  angle  of  elevation  of  the  top  of  the  cliff  is  56°  18'. 
What  is  the  height  of  the  cliff? 

62.  From  the  top  of  a  monument  the  angle  of  depression  of 
a  point  on  the  ground,  on   the  same  level  as  the  foot  of  the 
monument,  is  43°  41'.     The  point  is  found  by  measurement  to 
be  128.29  ft.  distant  from  the  foot  of  the  monument.     What 
is  the  height  of  the  monument? 

63.  From  the  top  of  a  hill  304  ft.  9  in.  in  height  the  angle 
of  depression  of  an  object  on  the  ground  is  40°  37'.     What  is 
the  distance  of  the  object  from  a  point  directly  below  the  point 
of  observation  and  on  the  same  level  with  the  object? 

64.  What  is  the  height  of  a  tree  that  casts  a  shadow  42.6  ft. 
long  when  the  angle  of  elevation  of  the  sun  is  60°  11'  ? 


THE   RIGHT   TRIANGLE  47 

65.  What  must  be  the  length  of  a  ladder  set  at  an  angle  of 
71°  14'  with  the  ground  to  reach  a  window  21.88  ft.  high  ? 

66.  To  find  the  width  of  a  river  a  point  P  is  selected  on 
one  bank,  and  a  distance  of  138.2  ft.  is  measured  parallel  to 
the  course  of  the  river  from  the  given  point  P  to  a  point  Q. 
Directly  opposite  (),  on  the  other  side  of  the  river,  is  the  point 
#,  and  the  angle  SPQ  is  found  to  be  66°  11'.     What  is  the 
width  of  the  river  ? 

67.  A  guy  rope  49.11  ft.  long  is  attached  to  the  top  of  a 
mast,  and   makes  an  angle  of  50°  56'  with  the   level   of   the 
ground.     What  is  the  height  of  the  mast  ? 

68.  The  top  of  a  flag  pole,  broken  by  the  wind,  falls  so  that 
it  touches  the  ground  at  a  distance  of  19. 73  ft.  from  the  foot  of 
the  pole,  and  is  inclined  to  the  ground  at  an  angle  of  65°  40'. 
What  is  the  height  of  the  portion  that  remains  standing,  and 
what  was  the  total  height  of  the  pole? 

69.  What  is  the  angle  of  elevation  of  an  inclined  plane  that 
rises  26  ft.  in  a  horizontal  distance  of  31.9  ft.  ? 

70.  A  man  walking  on  a  level  plain  toward  a  tower  observes 
that  at  a  certain  point  the  angle  of  elevation  of  the  top  of  the 
tower  is  30° ;  on  walking  300  ft.  directly  toward  the  tower  the 
angle  of   elevation  of   the  top  is  found   to  be  60°.     What  is 
the  height  of  the  tower  ? 

SOLUTION.     Let  x  =  the  height  of  the  tower  and  y  =  the  distance  from 
the  second  point  of  observation  to  the  foot  of  the  tower. 

From  the  triangle  A  CD     — =  tan  30°  =  —  , 

300  +  y  V3 

.-.  y  =  V3x  -  300; 
from  the  triangle  BCD  -  =  tan  60°  =  V3, 

y 


equating  these  values  of  y,  we  have 

V3z-300=  -*-, 

Vo 


300        B    y 


2  x  =  300  x  1.732, 

x  =  259.8. 


rd 

1    "     "'  ;   £   ' 


48 


PLANE   TRIGONOMETRY 


!  •> 


NOTE.  In  solving  problem  70  natural  functions  have  been  employed. 
On  p.  156  a  method  will  be  given  by  means  of  which  problems  of  this  kind 
can  be  solved  by  the  use  of  logarithms.  In  the  following  problems  it  is 
recommended  that  natural  functions  be  employed. 

v>  71.  At  a  point  on  a  level  plain  the  angle  of  elevation  of  the 
top  of  a  church  spire  is  45°,  and  at  a  point  50  ft.  nearer,  and  in 
the  same  straight  line  with  the  first  point  and  the  church,  the 
corresponding  angle  of  elevation  is  60°.  What  is  the  height  of 
the  spire? 

^  72.  From  the  top  of  a  cliff  150  ft.  high  the  angles  of  depres- 
sion of  the  top  and  bottom  of  a  tower  are  30°  and  GO0,  respec- 
tively. What  is  the  height  of  the  tower? 

73.  The  angles  of  elevation  of  the  top  of  a  tower,  taken  at 
two  points  268  ft.  apart  and  in  the  same  straight  line  with  the 
tower,  are  21°  14'  and  53°  4G',  respectively.     What  is  the  height 
of  the  tower? 

74.  At  the  foot  of  a  mountain  the  angle  of  elevation  of  the 
summit  is  45° ;  one  mile  up  the  slope  of  the  mountain,  which 
rises  at  an  inclination  of  30°,  the  angle  of  elevation  of  the  sum- 
mit is  60°.     What  is  the  height  of  the  mountain  ? 

75.  At  a  certain  point  south  of  a  tower  the  angle  of  eleva- 
tion of  the  top  of  the  tower  is  60°,  and  at  a  point  300  ft.  east 
of  the  point  the  corresponding  angle  of  elevation  is  30°.     What 

is  the  height  of  the  tower  ? 

30.   The   isosceles  triangle.     The 

perpendicular  from  the  vertex,  (7,  of 
an  isosceles  triangle  to  the  base 
divides  the  triangle  into  two  equal 
right  triangles. 

Any  two  parts  of  either  of  these 
right  triangles  being  given,  one  or 
both  of  which  are  sides,  the  right 

triangle  can  be  completely  determined.    Therefore  the  isosceles 

triangle  also  can  be  completely  determined. 

Denoting  the  base  of   the   isosceles   triangle  by  <?,  and  the 

altitude  by  h,  the  area,  A,  is  given  by  the  formula 

A  =  I  ch.  (1) 


- 


THE   RIGHT   TRIANGLE 


49 


31.  The  regular  polygon.  A  regular  polygon  is  divided  into 
equal  isosceles  triangles  by  lines  drawn  from  the  center  to  the 
vertices  of  the  polygon.  Each  of 
-the  isosceles  triangles  is  divided 
into  two  equal  right  triangles  by 
the  apothem  of  the  polygon. 

Any  side  of  either  of  these  right 
triangles  being  given,  the  polygon 
can  be  completely  determined  if 
the  number  of  sides  is  known. 

For  the  angles  at  the  center  of 
the  polygon  can  be  found  when 
the  number  of  sides,  n,  is  known,  by  dividing  360°  by  n. 
Taking  one  half  of  this  angle  as  one  of  the  acute  angles  of  the 
right  triangle,  and  combining  it  with  the  given  side,  we  have 
at  our  disposal  two  parts  of  a  right  triangle,  one  of  which  is  a 
side.  The  remaining  parts  can  then  be  found  by  the  methods 
already  given  for  the  solution  of  right  triangles. 

Denoting  the  perimeter  of  the  polygon  by  p  and  the  apothem 
by  h<  the  area  of  the  polygon  can  be  found  by  the  following 


formula : 


A   = 


(1) 


It  should  be  remembered  that  the  legs  of  the  isosceles  tri- 
angles are  radii  of  the  circumscribed  circle,  and  the  apothem 
is  the  radius'  of  the  inscribed  circle  of  the  polygon. 


EXERCISE  IX 

Solve    the    following    isosceles    triangles,    finding   the   part 
indicated  in  each  case : 

-jC     1.    Given    c=83.2,                      h  =  56.9;  find  C. 

2.  Given   c=  92.56,                    7i  =  59.72;  find  C. 

3.  Given    c=252.64,                 0=  62°  28'  36";  find  a. 

4.  Given  <7=  142°  27'  44",        a  =  92.452  ;  find  c. 

5.  Given  C=  102°  44'  42",        h  =  92.96 ;  find  a. 

6.  Given    c  =  85.32,                    A  =  49.84;  find  A. 
y-  7.    Given    c  =  136.48,                  A  =  60.51;  find  a. 

•L     8.    Given   A  =  1426.3,                  «=  2291.2;  find  A. 

CON  A  NT'S  TRin  — 4. 


50  PLANE   TRIGONOMETRY 

9.   Find  the  value  of  A  in  terms  of  a  and  0. 

10.  Find  the  value  of  A  in  terms  of  a  and  A. 

11.  Find  the  value  of  A  in  terms  of  h  and  A. 

Solve  the  following  regular  polygons,  having  given : 
~^12.   n«=10,          c='3.  14.   rc  =  6,  tf  =  12. 

13.   n=S,  h  =  2.  15.   n  =  20,         a  =  10. 

— •   16.    What  is  the  area  of  a  regular  octagon  formed  by  cut- 
ting away  the  corners  of  a  square  whose  side  is  6? 

— *  17.    What  is  the   area  of  a  circle  inscribed  in  an  equilat- 
eral triangle  whose  side  is  20? 

*^-    18.    What  is  the  area  of  a  regular  polygon  of  18  sides  if 
the  radius  of  the  circumscribed  circle  is  2? 

^     19.    One  of  the  diagonals  of  a  regular  pentagon  is  12.15. 

What  is  the  area  of  the  pentagon?  a*|.  0|ri>*" 

20.    Compute  the  area  of  a  regular  heptagon  if  the  length 

of  one  of  its  sides  is  13.88. 
v     21.    The    radius    of    the    circumscribed   circle  of   a   regular 

dodecagon  is  27.     What  is  the  area? 


CHAPTER   V 

THE   APPLICATION  OF  ALGEBRAIC  SIGNS  TO  TRIGO- 
NOMETRY 

32.  In    the   preceding  work   no  attempt  has  been  made  to 
apply  the  definitions  of  any  of  the  trigonometric  functions  to 
any  except  positive  acute  angles. 

These  definitions  will  now  be  extended  so  as  to  apply  to 
negative  as  well  as  to  positive  angles,  and  to  angles  of  any 
magnitude  whatever. 

33.  The  coordinate  axes.     The  location  of  a  point  or  a  line 
lying  in  a  given  plane  is  often  described  by  referring  it  to  two 
intersecting  straight  lines  in  that  plane,  called  coordinate  axes. 
These  lines  are  usually  drawn  perpendicular  to  each  other. 

Let  the  two  lines  XX'  and  YY'  intersect  at  right  angles. 
Then  the  plane  of  these  lines  is  divided  into  four  quadrants, 
designated  as  the  first,  second,  third,  and  fourth  quadrants, 
respectively.  These  quadrants  are  numbered  as  indicated  in 
the  figure. 

34.  Coordinates  of  a  point  in  a  plane.     The  location  of  any 
point  in  the  plane  determined  by  the  axes  XX'  and  YY'  is 
described  by  means  of  its  perpendicular  distances  from  these 
axes.  Y 

The  distance  of  a  point  from 
YY'  measured  along  a  line  parallel 
to  XX'  is  called  the  abscissa  of 
the  point ;  and  the  distance  of  a 

point  from  XX',  measured  on  a    X- 

line  parallel  to  YY'  is  called  the 
ordinate  of  the  point. 

The  abscissa  of  a  point  is  usu- 
ally designated  by  the  letter  x,  y' 

51 


52 


PLANE   TRIGONOMETRY 


and  the  ordinate  by  the  letter  y.     These  two  distances,  taken 
together,  are  called  the  coordinates  of  the  point. 

The  line  XX'  is  called  the  axis  of  abscissas,  and  the  line 
YY'  is  called  the  axis  of  ordinates.  These  axes  are,  for  the 
sake  of  brevity,  often  called  the  #-axis  and  the  #-axis,  respec- 
tively. Their  point  of  intersection,  0,  is  called  the  origin. 

Any  abscissa  measured  to  the  right  of  YY'  is  considered 
positive,  and  any  abscissa  measured  to  the  left  of  YY'  is  con- 
sidered negative. 

Any  ordinate  measured  above  XX'  is  considered  positive,  and 
any  ordinate  measured  below  XX'  is  considered  negative. 

The  coordinates  of  a  point  determine  its  position  completely. 
For  example,  if  the  point  A  is  4  units  from  YY1  and  6  units 

from  XX',  its  position  can  be 
located  as  follows :  measure  off 
on  XX'  a  distance  equal  to  4 
units,  and  through  the  point 
thus  found  draw  a  line  parallel 
to  YY'.,  Also,  measure  off  on 
YY1  a  distance  equal  to  6  units, 
and  through  the  point  thus  deter- 
mined, draw  a  line  parallel  to 
XX'.  The  intersection,  A,  of 
these  two  lines  is  the  required 


Y 


12 


point.  The  abscissa  of  A  is  4,  and  its  ordinate  is  6,  and  this 
point,  whose  location  is  given  by  means  of  its  coordinates,  is 
called  the  point  (4,  6). 

The  point  B,  located  in  a  similar  manner,  has  for  its  coordi- 
nates x  =  —  3  and  y  =  4  ;  and  this  point  B  is  called  the  point 
(—3,  4).  The  point  0 is  called  the  point  (—4,  —  5);  and  the 
point  D  is  called  the  point  (6,  —  3).  In  a  similar  manner  we 
can  locate  any  other  point  (#,  5),  where  a  and  b  are  any  real 
quantities  whatever,  either  positive  or  negative. 

35.  Trigonometric  functions  of  any  angle.  Let  the  line  OA 
(p.  53)  start  from  OX  and  revolve  in  a  positive  direction  until 
it  occupies  a  position  in  any  one  of  the  four  quadrants.  From 
any  point  P  in  the  revolving  line  draw  a  perpendicular  PM  to 
the  axis  of  abscissas,  XX' .  In  each  of  the  four  figures  we  have 


THE   APPLICATION   OF    ALGEBRAIC    SIGNS 


53 


CM=  x  and  MP  =  y.  Let  the  distance  OP  =  r.  The  trigono- 
metric functions  of  the  angle  XOA,  which  may  be  represented 
by  0,  are  then,  for  all  positions  of  OA,  defined  as  follows : 


M 


x 


X        0 


X 


x     x 


sill  0  = 


ordinate 


revolving  line      r 


/j  abscissa  x 

cos  u  =  —      - — : — =  -•> 

revolving  line      r 


=  = 

abscissa      x 

,  a      abscissa      x 
cot  u  =  -        —  =  -•> 
ordmate      y 

a      revolving  line      r 

sec  6  =  -  —  =  -•> 

x 


abscissa 


esc 


a  __  revolving  line  _  r 
ordinate  y 


54  PLANE   TRIGONOMETRY 

The  functions  vers  0  and  covers  0  are  denned  in  a  manner 
similar  to  that  employed  in  the  case  of  the  right  triangle,  as 
follows:  Ters  0=1  -cos*, 

covers  6  =  1  —  sin  6. 

NOTE.  In  the  case  of  cot  0°,  esc  0°,  tan  90°,  sec  90°,  cot  180°,  esc  180°,  tan 
270°,  sec  270°,  cot  360°  and  esc  360°,  these  definitions  fail.  For,  taking  as  an 
illustration  the  tangent  of  90°,  we  have  in  that  case  a  fraction  whose  numera- 
tor is  r  and  whose  denominator  is  0.  The  value  of  tan  90°  is,  then,  if  we 
attempt  to  use  the  above  definition,  given  by  this  fraction  whose  numerator 
is  r,  and  whose  denominator  is  0.  But  there  is  no  such  thing  as  division  by 
0,  hence,  according  to  the  definition  given,  the  symbol  tan  90°  has  no  mean- 
ing. This  and  other  similar  cases  will  be  discussed  later.  (See  pp.  57-63.) 

36.  In  a  manner  precisely  similar  to  that  employed  in  Art. 
16  it  can  be  proved  that,  for  any  value  whatever  of  0  the  fol- 
lowing relations  are  true  : 

sin2  6  +  cos2  6  =  1$  (1) 

sec2  6  =  1  +  tan2  6  ;  (2) 

esc2  0  =  1  +  cot2  6.  (3) 

Also,  from  the  definitions  of  the  functions,  the  following  rela- 
tions are  immediately  derived  : 

sin  6  =  -^-r,  .-.  sin  0  esc  6  =  1,  (4) 

CSC  U 
-^ 

cos0  =  -  -,  .-.  cos  6  sec  0  =  1,  (5) 

sec# 


-  - 

cot  u 


,  .-.  tan6cot0  =  l.  (6) 


Also,  since,          cos#  =  -,  . •.  a?  =  reos6,  (7) 


(8) 
(9) 


THE   APPLICATION   OF   ALGEBRAIC   SIGNS 


55 


37.  Signs  of  the  trigonometric  functions.  In  dealing  with 
the  functions  of  an  acute  angle  of  a  right  triangle  (Art.  14, 
p.  20),  no  attention  was  paid  to  the  question  of  positive 
or  negative  signs.  All  lines  employed  in  that  connection 
were  considered  positive;  hence  the  value  of  each  of  the 
functions  was  considered  positive.  But  in  dealing  with  the 
general  angle  we  have  to  consider  both  positive  and  nega- 
tive lines,  and  as  a  result  the  signs  of  the  functions  undergo 
certain  changes  as  the  revolving  line  passes  from  quadrant  to 
quadrant. 

First  Quadrant.  Assume  that  the  revolving  line  is  always 
positive,  and  let  it  occupy  any  position  in  the  first  quadrant. 

In  this  position  both  x  and  y  are  positive;  hence,  since  r  is 
also  positive,  both  numerator  and  denominator  are  positive  in 
the  case  of  each  of  the  functions.  Therefore  all  the  trigono- 
metric functions  are  positive  for  the  angle  in  the  first  quadrant. 


Y 


Second  Quadrant.  Let  the  revolving  line  occupy  any  posi- 
tion in  the  second  quadrant.  In  this  case  x  is  negative  and  y 
is  positive;  and  we  have  the  following  results: 

The  sine  is  a  fraction  whose  numerator  and  denominator  are 
both  positive ;  therefore  the  sine  of  an  angle  in  the  second 
quadrant  is  positive. 

The  cosine  is  a  fraction  whose  numerator  is  negative  and 
whose  denominator  is  positive;  therefore  the  cosine  of  an  angle 
in  the  second  quadrant  is  negative. 

The  tangent  is  a  fraction  whose  numerator  is  positive  and 
whose  denominator  is  negative;  therefore  the  tangent  of  an 
angle  in  the  second  quadrant  is  negative. 


56 


PLANE   TRIGONOMETRY 


The  cotangent  is  a  fraction  whose  numerator  is  negative  and 
whose  denominator  is  positive;  therefore  the  cotangent  of  an 
angle  in  the  second  quadrant  is  negative. 

The  secant  is  a  fraction  whose  numerator  is  positive  and 
whose  denominator  is  negative;  therefore  the  secant  of  an 
angle  in  the  second  quadrant  is  negative. 

The  cosecant  is  a  fraction  whose  numerator  and  denominator 
are  both  positive;  therefore  the  cosecant  of  an  angle  in  the 
second  quadrant  is  positive. 

Third  Quadrant.  Let  the  revolving  line  occupy  any  position 
in  the  third  quadrant.  In  this  case  both  x  and  y  are  negative; 
therefore  the  following  results  can  at  once  be  obtained: 

The  sine  is  negative. 
The  cosine  is  negative. 
The  tangent  is  positive. 
The  cotangent  is  positive. 
The  secant  is  negative. 
The  cosecant  is  negative. 


Fourth  Quadrant.  Let  the  revolving  line  occupy  any  position 
in  the  fourth  quadrant.  In  this  case  x  is  positive  and  y  is 
negative;  therefore  the  following  results  can  at  once  be 
obtained:  The  gine  .g  negative< 

The  cosine  is  positive. 
The  tangent  is  negative. 
The  cotangent  is  negative. 
The  secant  is  positive. 
The  cosecant  is  negative. 


THE    APPLICATION   OF   ALGEBRAIC    SIGNS 


57 


The  above    results   are    conveniently  grouped   together    by 
means  of  the  following  table: 


sine               + 

sine               -f- 

cosine 

cosine            + 

tangent         — 
cotangent     — 
secant 

tangent         + 
cotangent     -f 
secant           + 

cosecant        + 

cosecant       + 

Y 

sine 

sine 

cosine 

cosine            + 

tangent         + 
cotangent      + 
secant 

tangent 
cotangent     — 
secant           + 

cosecant 

cosecant       — 

Y' 

From  the  definitions  of  the  versed  sine  and  of  the  coversed 
sine  it  follows  that  these  two  functions  are  always  positive. 

38.  Changes  in  sign  and  magnitude  of  the  trigonometric  func- 
tions as  the  angle  increases  from  0°  to  360°. 

As  before,  we  assume  for  the  revolving  line  a  constant  length, 
r.  As  the  revolving  line  starts  from  its  initial  position  we 
have  x  =  r,  and  y  =  0.  As  the  angle  0,  which  is  generated  by 
the  revolution  of  this  line,  increases  from  0°  to  90°,  y  increases 
and  x  decreases;  and  when  OA  coincides  with  OY,  we  have 
x  =  0,  and  y  =  r.  Hence,  as  the  angle  increases  from  0°  to  90°, 
x  decreases  from  r  to  0,  and  y  increases  from  0  to  r. 

As  the  angle  increases  from  90°  to  180°,  x  decreases  —  in- 
creases numerically  —  from  0  to  —  r  and  y  decreases  from  r 
to  0. 

As  the  angle  increases  from  180°  to  270°,  x  increases  —  de- 
creases numerically  —  from  —  r  to  0  and  y  decreases — increases 
numerically  —  from  0  to  —  r. 

As  the  angle  increases  from  270°  to  360°,  x  increases  from  0 
to  r  and  y  increases  —  decreases  numerically  —  from  —  r  to  0. 

Inasmuch  as  all  changes  in  sign  and  magnitude  among  the 
trigonometric  functions  are  directly  dependent  on  the  changes 
just  noted,  the  following  results  are  now  obtained  without 
difficulty. 


58 


PLANE   TRIGONOMETRY 


X- 


X       X- 


Y 
Y 


Y 
Y 


39.  Sine.  As  the  angle  increases  from  0°  to  90°  the  numera- 
tor of  the  fraction  that  expresses  the  value  of  the  sine  increases 
from  0  to  r,  and  the  denominator  r  remains  constant.  Hence 
the  sine  increases  from  0  to  1.  As  the  angle  increases  still 
further,  the  numerator  begins  to  decrease,  the  denominator  still 
remaining  constant,  and  at  180°  the  numerator  becomes  0. 
Hence  as  the  angle  increases  from  90°  to  180°  the  sine  decreases 
from  1  to-  0.  As  the  revolving  line  enters  the  third  quadrant, 
y  becomes  negative  and  continues  to  decrease  algebraically, 
becoming  —  r  when  the  angle  equals  270°.  Hence  in  the  third 
quadrant  the  sine  is  negative,  and  as  the  angle  increases  from 
180°  to  270°  the  sine  decreases  from  0  to  -  1.  In  the  fourth 
quadrant  y  continues  negative ;  but  as  the  angle  increases  y 
increases  algebraically,  and  when  the  revolving  line  reaches  its 
original  position,  y  again  becomes  0.  Hence  as  the  angle 
increases  from  270°  to  360°  the  sine  is  negative,  and  increases 
from  —  1  to  0. 

Collecting  the  above  results  for  the  sake  of  convenience  we 
have  the  following  statement  : 


THE   APPLICATION   OF   ALGEBRAIC   SIGNS  59 

In  the  first  quadrant  the  sine  increases  from  0  to  1;  in  the 
second  it  decreases  from  1  to  0 ;  in  the  third  it  decreases  from 
0  to  —  1 ;  in  the  fourth  it  increases  from  —  1  to  0. 

40.  Cosine.     In  a  manner  similar  to  that  employed  in  the 
case  of  the  sine,  the  following  results  are  obtained: 

As  the  angle  increases  from  0°  to  90°  the  cosine  decreases 

from  -  to  -,  i.e.   from  1   to  0.      As  the  angle  increases  from 

r        r 

90°  to  180°  the  cosine  decreases  —  increases  numerically  —  from  - 

T 

to  ^^,  i.e.  from  0  to  —  1.  As  the  angle  increases  from  180° 
to  270°  the  cosine  increases  —  decreases  numerically  —  from' 

—  to  -,  i.e.  from  —  1  to  0.     As  the  angle  increases  from  270° 
r          r 

to  360°  the  cosine  increases  from  -  to  -,  i.e.  from  0  to  1. 

r        r 

41.  Tangent.     The  value  of  the  tangent  is  the  value  of  the 

fraction   ^  •     When  the  angle  is  very  small,  the  numerator  of 
x 

this  fraction  is  very  small,  and  the  denominator  is  very  nearly 
equal  to  r.  Hence  the  tangent  of  the  angle  is  very  small ;  or, 
as  it  is  commonly  expressed,  when  the  angle  equals  0°,  the 
tangent  of  the  angle  is  also  equal  to  0. 

As  the  angle  increases  the  numerator  y  increases  and  the 
denominator  x  decreases.  Hence  the  tangent  of  the  angle 
increases.  When  the  angle  is  nearly  90°,  the  numerator  is 
very  nearly  equal  to  r\  and  as  the  angle  approaches  90°  the 
value  of  the  numerator  continually  increases,  approaching  r  as 
its  limit.  At  the  same  time  the  value  of  the  denominator  con- 
tinually decreases,  approaching  0  as  its  limit.  Hence,  as  0 
approaches  90°  the  value  of  tan  0  can  be  made  to  exceed  any 
finite  number  previously  assigned,  no  matter  how  great  that 
number  may  be.  This  is  usually  expressed  by  saying  that 
when  the  angle  is  equal  to  90°,  the  tangent  of  the  angle  is  equal 
to  infinity.  Hence,  ~ 

In  the  first  quadrant  the  tangent  increases  from  -  to  -,  i.e. 
from  0  to  QO  . 

In  the  second  quadrant  the  denominator  x  becomes  negative 
while  the  numerator  y  remains  positive.  Hence  the  tangent 


60  PLANE   TRIGONOMETRY 

of  an  angle  in  the  second  quadrant  is  negative.  When  the 
angle  is  but  little  greater  than  90°,  the  numerator  is  very 
nearly  equal  to  r  and  the  denominator  is  very  small,  and  nega- 
tive. Therefore,  as  the  revolving  line  enters  the  second 
quadrant,  the  numerical  value  of  the  tangent  can  be  taken  to  be 
greater  than  any  negative  finite  limit  previously  assigned. 
That  is,  when  the  angle  is  in  the  second  quadrant  and  differs  from 
90°  by  an  amount  that  is  less  than  any  finite  number  assigned 
in  advance,  no  matter  how  small  that  number  may  be,  the 
tangent  of  the  angle  is  negative  and  is  numerically  greater  than 
any  finite  limit  assigned  in  advance.  To  express  this  we  shall 
'say  that  tan  90°  =  —  oo.  It  is  thus  seen  that  tan  90°  will  be 
called  equal  to  either  +00  or  —  oo  according  as  the  angle  is 
approaching  the  limit  90°  from  the  positive  direction,  or  as  the 
revolving  line  is  leaving  the  position  at  which  the  angle  equals 
90°  and  is  just  entering  the  second  quadrant.  As  the  angle 
increases,  the  numerator  decreases  and  the  denominator,  which 
is  negative,  increases  numerically.  Hence,  the  tangent  decreases 
numerically  —  increases  algebraically  —  and  when  the  angle 
becomes  equal  to  180°,  the  tangent  becomes  equal  to  0.  Hence, 

In  the  second  quadrant  the  tangent  increases  from  -  to  , 

i.e.  from  —  oo  to  0. 

In  the  third  quadrant  both  numerator  and  denominator  are 
negative.  Hence  the  tangent  is  positive.  The  numerator  in- 
creases numerically  from  0  to  —  r,  and  the  denominator  de- 
creases numerically  from  —  r  to  0.  Hence, 

In  the  third  quadrant  the  tangent  increases  from  — -  to  -^, 
i.e.  from  0  to  oo. 

In  the  fourth  quadrant  the  numerator  is  negative  and  the 
denominator  is  positive.  Hence  the  tangent  is  negative.  The 
numerator  decreases  numerically  from  —  r  to  0,  and  the  de- 
nominator increases  from  0  to  r.  Hence, 

In  the  fourth  quadrant  the  tangent  increases  from  -^—  to  — , 
i.e.  from  —  oo  to  0. 

The  same  restriction  is  to  be  observed  with  respect  to  the 
value  of  tan  270°  as  was  noted  in  connection  with  tan  90°. 
That  is,  if  the  angle  is  in  the  third  quadrant  and  is  approaching 
270°  as  its  limit,  the  tangent  of  the  angle  can  be  made  to  exceed 


THE   APPLICATION   OF   ALGEBRAIC   SIGNS  61 

in  magnitude  any  finite  positive  limit  previously  assigned.     If 

it  is  in  the  fourth  quadrant,  the  tangent  is  negative  and  can 

be  made  to  exceed  in   numerical  magnitude  any  finite  limit 

_previously  assigned.      For  this  reason  it  is  customary  to  say 

that  tan  270°  =  ±  oo  . 

42.  Cotangent.     The  value  of  the  cotangent  is  the  value  of 
the  fraction  -.     When  the  angle  is  very  small,  the  numerator 

y 

is  nearly  equal  to  r  and  the  denominator  is  nearly  equal  to  0. 
Hence  the  value  of  the  cotangent  of  0°  is  infinity.  Then, 
letting  the  angle  increase,  and  reasoning  in  the  same  manner 
as  in  the  case  of  the  tangent,  we  obtain  the  following 
results : 

In  the  first  quadrant  the  cotangent  is  positive  and  decreases 

T  0 

from  —  to  — ,  i.e.  from  GO  to  0. 
0        r 

In  the  second  quadrant  the  cotangent  is  negative  and  de- 

0         —  r 

creases  from  —  to  ,  i.e.  from  0  to  -co. 

r          0 

In  the  third  quadrant  the  cotangent  is  positive  and  decreases 

—  r          0 

from  to  ,  i.e.  from  oo  to  0. 

0  -r 

In  the  fourth  quadrant  the  cotangent  is  negative  and  de- 

0  T 

creases  from  — -  to  -,  i.e.  from  0  to  —  oo. 
-r        0 

Remarks  similar  to  those  made  in  connection  with  tan  90°  and 
tan  270°  apply  to  cot  0°,  cot  180°,  and  cot  360°. 

43.  Secant.     The  value  of   the   secant   is   the  value  of   the 

fraction  -.     The  numerator  remains  constant  for  all  positions 
x 

of  the  revolving  line,  while  the  denominator  varies.  When  the 
angle  is  very  small,  the  numerator  and  the  denominator  are 
approximately  equal.  Hence  the  secant  of  0°  is  equal  to  unity. 
As  the  angle  increases  the  denominator  x  decreases,  thus  caus- 
ing the  value  of  the  secant  to  increase.  When  the  angle  is 
nearly  equal  to  90°,  the  denominator  is  nearly  equal  to  0, 
and  approaches  0  as  its  limit.  Therefore  the  secant  can  be 


62  PLANE   TRIGONOMETRY 

made  to  exceed  any  finite  limit  previously  assigned.      We  shall 
express  this  by  saying  that  sec  90°  =  <x> .      Hence, 

As  the  angle  increases  from  0°  to  90°  the  secant  increases 

from  -  to  — ,  i.e.  from  +1  to  +00  . 
r        0 

When  the  revolving  line  enters  the  second  quadrant  the 
denominator  x  becomes  negative  and  begins  to  increase  numeri- 
cally —  decrease  algebraically  —  becoming  equal  to  —  r  when 
the  angle  becomes  180°.  Hence,  beginning  with  a  negative 
value  numerically  greater  than  any  finite  limit  assigned  in 
advance,  the  secant  increases  —  decreases  numerically  —  until  it 
reaches  the  value  —  1.  Hence, 

As  the  angle  increases  from  90°  to  180°  the  secant  increases 

T  T 

from  -^  to ,  i.e.  from  —  oo  to  —  1. 

0         —  r 

In  the  third  quadrant  the  denominator  continues  negative, 
but  begins  to  decrease  —  increase  numerically  —  as  soon  as  the 
revolving  line  enters  the  quadrant.  At  270°  the  denominator 
becomes  0.  Hence, 

As  the  angle  increases  from  180°  to  270°  the  secant  decreases 

from   — —  to  -,  i.e.  from  —  1  to  —  oo. 
—  r        0 

In  the  fourth  quadrant  the  denominator  again  becomes  posi- 
tive, and  increases  from  0  to  r  as  the  angle  increases  from  270° 
to  360°,  returning  to  its  original  value  when  the  revolving  line 
completes  one  entire  revolution.  Hence, 

As  the  angle  increases  from  270°  to  360°  the  secant  decreases 

from  -^  to  -,  i.e.  from  oo  to  1. 
0        r 

The  same  restriction  is  to  be  observed  with  respect  to  the 
value  of  sec  270°  as  was  noted  in  connection  with  sec  90°. 
That  is,  if  the  angle  is  in  the  third  quadrant  and  is  approaching 
270°  as  its  limit,  the  secant  of  the  angle  can  be  made  to  exceed 
in  numerical  magnitude  any  finite  negative  limit  assigned  in 
advance.  If  the  angle  is  in  the  fourth  quadrant,  the  secant  is 
positive,  and  can  be  made  to  exceed  in  magnitude  any  finite 
positive  limit  assigned  in  advance.  We  shall  express  this  by 
saying  that  sec  270°  =  ±cc  . 


THE   APPLICATION    OF   ALGEBRAIC   SIGNS 


63 


44.  Cosecant.  The  value  of  the  cosecant  is  the  value  of  the 
fraction  -.  Remembering  that  the  numerator  remains  con- 

y 

stant,  and  tracing  out  the  changes  in  sign  and  magnitude  of 
the  denominator,  as  in  the  case  of  the  secant,  we  obtain  the  fol- 
lowing results : 

As  the  angle  increases  from  0°  to  90°  the  cosecant  decreases 

from  -£  to  -,  i.e.  from  oo  to  1. 
0        r 

As  the  angle  increases  from  90°  to  180°  the  cosecant  increases 

7*  7* 

from  -  to  -,  i.e.  from  1  to  oo. 
r       0 

As  the  angle  increases  from  180°  to  270°  the  cosecant  increases 

y  v* 

from  -  to ,  i.e.  from  —  oo  to  —  1. 

0        —  r 

As  the  angle  increases  from  270°  to  360°  the  cosecant  decreases 

'/*  '/* 

from  to  -,  i.e.  from  —  1  to  —  oo. 

—  r       0 

Remarks  similar  to  those  made  in  connection  with  sec  90° 
and  sec  270°  apply  to  esc  0°,  esc  180°,  and  esc  360°. 

The  changes  that  take  place  in  the  sign  and  magnitude  of 
the  different  trigonometric  functions  are  conveniently  grouped 
together  in  the  following  table : 

r 

FIRST  QUADRANT 
increases  from  0  to  1 


SECOND  QUADRANT 

sine            decreases  from  1  to  0 
cosine         decreases  from  0  to  —  1 
tangent      increases  from  —  GO  to  0 
cotangent  decreases  from  0  to  —  oo 
secant        increases  from  —  x  to  —  1 
cosecant     increases  from  1  to  oo 
X' 


sine 

cosine  decreases  from  1  to  0 
tangent  increases  from  0  to  oo 
cotangent  decreases  from  oo  to  0 
secant  increases  from  1  to  oo 
cosecant  decreases  from  oo  to  1 


THIRD  QUADRANT 

sine  decreases  from  0  to-  —  1 

cosine         increases  from  —  1  to  0 
tangent     increases  from  0  to  oo 
cotangent  decreases  from  oo  to  0 
secant        decreases  from  —  1  to  —  oo 
cosecant    increases  from  -co  to  —1 


FOURTH  QUADRANT 

sine  increases  from  —  1  to  0 

cosine        increases  from  0  to  1 
tangent      increases  from  —  oo  to  0 
cotangent  decreases  from  0  to  —  oo 
secant        decreases  from  oo  to  1 
cosecant    decreases  from  —1  to  -co 


T' 


45.  After  the  changes  in  sign  and  magnitude  have  been 
obtained  for  the  first  three  functions,  the  corresponding  changes 
for  the  last  three  can  be  found  by  remembering  that  the 


64 


PLANE   TRIGONOMETRY 


cotangent,  secant,  and  cosecant  are  the  reciprocals  of  the 
tangent,  the  cosine,  and  the  sine  respectively.  The  student 
should  verify  the  above  results  by  obtaining  them  in  this 
manner  also. 

In  connection  with  the  general  definitions  of  the  trigonometric 
functions  given  on  p.  53,  it  was  noted  that  these  definitions 
failed  in  the  case  of  certain  functions  for  certain  values  of  the 
angle.  These  cases  have  been  explained  in  some  detail  in  Arts. 
41-44,  and  we  now  have  definitions  of  the  tangent,  the  cotan- 
gent, the  secant,  and  the  cosecant  of  any  angle  from  0°  to  360° 
inclusive ;  and  hence,  by  the  usual  considerations,  Arts.  50-57, 
definitions  of  these  functions  for  any  angle  whatever. 

In  order  that  the  relations  between  tan  90°  and  cot  90°,  tan 
270°  and  cot  270°,  sec  90°  and  cos  90°,  etc.,  may  be  the  same  as 
that  between  the  same  functions  in  the  case  of  other  angles  we 

shall  say  that  —  =  0,  and  -  =  oo  .     But  the  student  is  cautioned 
oo  0 

that  "oo  "  is  not  a  number  in  the  usual  sense  of  the  word,  and 
that  these  two  equations  are  not  to  be  taken  literally.  They 
are  used  merely  for  the  sake  of  expressing  concisely  the  result 
of  a  definite  limiting  process,  a  process  much  more  complicated 
than  that  of  ordinary  division. 

46.  Geometrical  representation  of  the  trigonometric  functions. 
The  trigonometric  functions  are  pure  numbers,  the  value  in  each 
case  being  a  ratio  between  two  given  magnitudes.  These 
magnitudes  are  represented  by  lines,  and  if  the  length  of 
the  revolving  line  is  properly  chosen,  it  is  possible  to  represent 
the  values  of  the  functions  themselves  by  lines. 

Let  the  revolving  line  be  the 
radius  of  a  circle,  and  let  its  value 
be  assumed  to  be  unity. 

The  sine  of  the  angle  AOB  is 

CD 

— -.     But  since  OD=\,  we  may 


OD 

say 


CD      CD 


Similarly, 


THE   APPLICATION   OF   ALGEBRAIC   SIGNS 


65 


vers  <9  =  1  -  cos  0  =  (L4  -  00=  AC, 
covers  0  =  1  -  sin  (9  =  OG  -  OE  =  GE. 

For   an  angle  of  the  second  quadrant   the    so-called   "line 
values"  of  the  trigonometric  functions  are  obtained  as  follows: 


vers  0  =  1  -  cos  0  =  0.4  -  00=  OA+00=  CA, 
covers  0  =  1  -  sin  0  =  OG-  OE=E&. 

The  change  in  sign  when  00  is  replaced  by  00  in  obtain 
ing  the    value  of   the  versed    sine  should  be  noted  carefully. 
CONANT'S  TRIG.  —  5 


66 


PLANK    TRIGONOMETRY 


For  angles  of  the  third  and  fourth  quadrants  the  line  values 
are  obtained  in  a  manner  similar  to  that  employed  in  connec- 
tion with  angles  of  the  first  two  quadrants.  The  figures  are 
lettered  so  that  the  following  values  hold  for  both : 


oc    an 

=      = 


0(J      OA 


an 


esc 


OI>      OH      OH 

=  —  -  =  -  —  —  OH, 

CD    oa      i 


vers  0=1-  cos  0=OA-  OC=  OA, 
covers  0=1  -  sin  0  =  GO-  -  OE=Ea. 

The  signs  of  the  trigonometric  functions  when  used  as  lines 
are,  of  course,  the  same  as  when  they  are  used  as  ratios.  It 
will  be  noticed  that  when  the  line  that  represents  the  sine 
extends  upward  from  the  axis  of  abscissas,  or  horizontal  di- 
ameter, the  sine  is  positive  ;  when  it  extends  downward,  the 


THE    APPLICATION   OF  ALGEBRAIC   SIGNS  67 

sine  is  negative.  The  cosine  is  positive  when  the  line  that 
represents  its  value  extends  toward  the  right  from  the  origin, 
negative  when  it  extends  toward  the  left.  The  tangent  is 
positive  when  its  line  extends  upward  from  the  axis  of  abscissas, 
or  horizontal  diameter,  negative  when  it  extends  downward. 
The  cotangent  is  positive  when  its  line  extends  toward  the 
right  from  the  axis  of  ordinates,  or  vertical  diameter,  negative 
when  it  extends  toward  the  left.  The  secant  and  the  cosecant 
are  positive  when  their  respective  lines  extend  in  the  same 
direction  from  the  origin  as  the  revolving  line,  negative  when 
they  extend  in  an  opposite  direction.  The  versed  sine  is  con- 
sidered as  extending  toward  the  right  from  the  foot  of  the  sine, 
and  the  coversed  sine  upward  from  the  foot  of  the  perpen- 
dicular dropped  from  the  extremity  of  the  revolving  line  to  the 
vertical  diameter.  Both  are  always  positive. 

The  trigonometric  functions  were  originally  used  as  lines ; 
and  the  numerical  value  was,  in  each  case,  the  length  of  the 
line  in  terms  of  the  revolving  line,  or  the  radius  of  the  circle, 
taken  as  a  unit.  There  are  certain  advantages  connected  with 
the  use  of  these  line  values,  but  for  general  purposes  the  ratios 
are  so  much  more  convenient  than  the  line  values  that  they 
have  now  come  into  almost  universal  use. 

47.  Limiting  values  of  the  trigonometric  functions.  In  dis- 
cussing the  variation  in  the  values  of  the  different  functions 
the  following  limits  were  found.  In  the  case  of  the  sine  the 
positive  limit  was  1,  and  the  negative  limit  was  —  1.  For  the 
cosine  also  these  limits  were  -f  1  and  --  1  respectively.  For 
the  tangent  and  the  cotangent  the  limits  were  +00  and  -co. 
For  the  secant  and  the  cosecant  it  was  found  that  the  positive 
values  that  these  functions  could  take  were  comprehended 
between  +  1  and  +  oo,  and  the  negative  values  between  —  1 
and  —  co.  Hence,  we  can  make  the  following  definite  state- 
ment respecting  the  limits  between  which  the  different  func- 
tions can  vary : 

The  sine  can  take  any  value  between  -f- 1  and  —  1. 

The  cosine  can  take  any  value  between  +  1  and  —1. 

The  tangent  can  take  any  value  between  -f  oo  and  —  oo. 

The  cotangent  can  take  any  value  between  +  oo  and  —  GO. 


68 


PLANE   TRIGONOMETRY 


The  secant  can  take  any  value  between  + 1  and  +  oo,  and 
between  —  1  and  —  oo. 

The  cosecant  can  take  any  value  between  -f-  1  and  +  oo,  and 
between  —  1  and  —  oo. 

From  the  definitions  of  the  versed  sine  and  the  coversed  sine 
it  follows  that  each  of  these  functions  can  take  any  value  be- 
tween 0  and  -f  2. 

48.  Graphs  of  the  trigonometric  functions.  The  graphs  of 
the  trigonometric  functions  can  be  plotted  in  the  ordinary 
manner  if  the  values  of  the  angles  are  taken  as  ordinates  and 
the  corresponding  values  of  the  functions  as  abscissas. 

Sine.  For  the  sine  we  form  the  following  table  of  values 
from  the  equation  y  ~  sjn  Xt 

In  this  table  the  values  of  the  sine  are,  for  convenience  in 
plotting,  given  decimally,  instead  of  in  the  ordinary  common 
fractious. 


/scf 


30  00  QO  120  f 50 


-A' 


X 

y 

0° 

0 

30° 

.5 

45° 

.71 

60° 

.87 

90° 

1 

120° 

.87 

135° 

.71 

150° 

.5 

180° 

0 

225° 

-  .71 

270° 

-  1 

315° 

-  .71 

360° 

0 

415° 

.71 

450° 

1 

495° 

.71 

etc. 

etc. 

Continuing  this  table,  and  plotting  the  points  thus  deter- 
mined, we  find  that  the  graph  is  a  curve  consisting  of  an 
infinite  number  of  waves  like  those  in  the  figure.  By  using 
negative  values  of  the  angle  we  obtain  similar  waves  at  the  left 
of  the  origin.  The  curve  is  called  the  sine  curve,  or  sinusoid. 


THE   APPLICATION   OF  ALGEBRAIC   SIGNS 


69 


Cosine.     The  graph  of  the  equation 

y  =  cos  x 

is  found  in  a  similar  manner.  Forming  a  table  of  values,  and 
plotting  the  points  determined  by  these  values,  we  find  that 
the  cosine  curve  has  the  following  form. 


90 


30  60 


Y 

Tangent.     The  table  of  values  for  x  and  y  formed  from  the 

e(luation  #  =  tan* 

is  as  follows. 


X 

y 

0° 

0 

30° 

.58 

45° 

1 

60° 

1.73 

90° 

00 

120° 

-  1.73 

135° 

-  1 

150° 

-  .58 

180° 

0 

210° 

.58 

225° 

1 

240° 

1.73 

270° 

CO 

300° 

-  1.73 

315° 

-1 

330° 

-.58 

360° 

0 

390° 

.58 

etc. 

etc. 

-270° 


-96 


30°  60  90 


I 2O  150' 


I  /ISO       270 


70 


PLANE   TRIGONOMETRY 


Continuing  the  table,  and  plotting  the  points  determined  by 
the  values  thus  found,  we  obtain  the  tangent  curve,  which  con- 
sists of  an  infinite  number  of  branches,  each  like  one  of  those 
in  the  figure.  Negative  values  of  the  angle  give  an  infinite 
number  of  like  branches  at  the  left  of  the  origin. 

Cotangent.     The  graph  of  the  equation 

y  =  cot  x 

is  similar  to  that  of  y  =  tan  #,  except  that  the  points  where  the 
different  branches  cross  the  #-axis  are  90°  to  the  right  of  those 
where  the  tangent  curve  branches  cross,  and  the  curvature  is 
toward  the  right  instead  of  toward  the  left.  The  form  of  the 
graph  is  shown  in  the  following  figure. 


X 

y 

0° 

CO 

30° 

1.73 

45° 

1 

60° 

.58 

90° 

0 

120° 

-.58 

135° 

-  1 

150° 

-  1.73 

180° 

—    CO 

210° 

1.73 

225° 

1 

240° 

.58 

270° 

0 

300° 

-.58 

315° 

-1 

330° 

-1.73 

360° 

CO 

390° 

1.73 

etc. 

etc. 

Secant.     The  table  of  values  for  the  equation 

y  —  sec  x 

can  readily  be  found  if  it  is  remembered  that  sec#  is  the  recip- 
rocal of  cosx.  The  graph  has  the  form  shown  in  the  first  figure 
on  p.  71. 

Cosecant.    The  graph  of  the  cosecant  is  similar  in  form  to 
that  of   the    secant,  but    the    relative   position  of  the  various 


THE   APPLICATION   OF   ALGEBRAIC   SIGNS 
Y 


71 


-36O      -i 


branches  with  respect  to  the  #-axis  is  different.     The  graph  is 
shown  in  the  following  figure. 


49.  Periods  of  the  trigonometric  functions.  In  considering  the 
changes  in  value  through  which  the  functions  pass  as  the  angle 
increases,  it  is  seen  that  the  sine,  for  example,  takes  all  its  pos- 
sible values,  in  both  increasing  and  decreasing  order  of  change, 
while  the  angle  is  increasing  from  0°  to  360°.  As  the  angle 
increases  from  360°  to  720°  the  values  of  the  sine  which  were 
obtained  in  the  first  360°  are  repeated,  this  repetition  of  values 
occurring  in  the  original  order.  The  same  cycle  of  values  will 
again  occur  in  the  next  360°,  and  so  on,  for  each  complete 
revolution  of  the  generating,  or  revolving  line.  The  angle 
formed  by  the  generating  line  while  this  regular  recurrence 
of  values  takes  place  is  called  the  period  of  the  sine;  and  in 
accordance  with  this  result  we  may  say  that 

The  period  of  the  sine  is  360°,  or  2  IT. 

A  similar  course  of  reasoning  shows  us  that  360°  is  also  the 
period  of  the  cosine,  of  the  secant,  and  of  the  cosecant. 


72  PLANE   TRIGONOMETRY 

The  values  of  the  tangent  repeat  themselves  completely  with 
each  increase  of  180°  in  the  angle.  Hence, 

The  period  of  the  tangent  is  180°,  or  IT. 

The  period  of  the  cotangent  is  the  same  as  the  period  of  the 
tangent. 

EXERCISE  X 

1.  Trace  the  changes  in  sign  and  magnitude  of  sin  6  as  0 
varies  from  -  ^  to  -  ^  ;    from  -  270°  to  -  450°. 

2.  Trace  the  changes  in  sign  and  magnitude  of  cos  A  as  A 
varies  from  —  TT  to  —  2  TT. 

3.  Trace  the  changes  in  sign  and  magnitude  of  tan  A  as  A 
varies  from  —  180°  to  —540°. 

4.  Trace  the  changes  in  sign  and  magnitude  of  sec  A  as  A 
varies  from  -  90°  to  -270°. 

Find  the  value  of  each  of  the  following: 

5.  sin  6  +  cos  6  when  6  =  60°. 

6.  sin2  6  +  2  cos  0  when  0  =  45°. 

7.  sin  A  4-  tan  A  when  A  =  135°. 

8.  sin  60°  +  tan  240°. 

9.  cos  0°  cos  30°  +  tan  135°  cot  315°. 

10.  cos  0°  tan  60°  -  sec2  30°  cot  225°. 

11.  2  sin  90°  sec2  30°  +  cos  180°  tan  315°. 

12.  2sec27rcos00  +  3sin3^-csc^. 

2  2t 

13.  COSTT  tan  !L  -  sec2  11^  tan2  ii 

464 

14.  For  which  of  the  following  values  of  6  is  sin  6  —  cos  6 
positive  and  for  which  is  it  negative? 

(9  =  0°;  (9=60°;  0  =  120° ;  (9=210°;  (9=240°;  (9=300°; 
6  =  330°. 


THE   APPLICATION   OF   ALGEBRAIC   SIGNS  73 

15.  For  which  of  the  following  values  of  0  is  sin  0  -f  cos  0 
positive  and  for  which  is  it  negative? 

(9  =  135°;  (9  =  210°;  0=300°;  0=315°;  0=330°. 

16.  Prove  that  sec6  0  -  tan6  0=3  sec2  0  tan2  0  +  1. 

9          79 

17.  If  cos  0  =  a0  ~  _0,  find  sin  0  and  tan  0. 

a2  4-  62 


18.  If  tan  0  =     a  +     a,  find  cos  0  and  sin  0. 

2a  +  1 

19.  Prove  the  equation  sin0  =  #  +  -  impossible  for  all  real 
values  of  x. 

20.  Prove  the  equation  sec2  0  =        ^   .    impossible  unless 


CHAPTER   VI 
TRIGONOMETRIC    FUNCTIONS   OF   ANY   ANGLE 

50.  Functions  of  an  angle  —  9  in  terms  of  functions  of  6. 
Let  the  revolving  line  OA  generate  an  angle  0,  of  any  mag- 
nitude. The  final  position  of  OA  is,  then,  in  any  one  of  the 
four  quadrants,  as  shown  in  the  figures.  Also,  let  the  line 
OA'  generate  an  angle  —  0,  equal  in  magnitude  to  the  positive 
angle  0,  generated  by  OA. 

Take  OB=  OB1,  and  from  B  and  B'  draw  perpendiculars 
BC,  B'C',  to  XX'.  Then  are  the  triangles  OBC,  OB'O',  equal 
geometrically,  since  they  are  right  triangles  having  the  hypote- 
nuse and  an  acute  angle  of  one  equal  respectively  to  the  hypote- 
nuse and  an  acute  angle  of  the  other.  Hence  the  points  6Y,  6Y/, 
coincide,  BO=B'Q'<  and  00=  0' C' . 


TRIGONOMETRIC   FUNCTIONS   OF   ANY   ANGLK  75 


For   convenience,   let  OB  =  r,    OB'  =  rf,  BC  =  y,   B'C'=y', 
00  =  x,  OC'  =  z'  -,  then  for  each  of  the  four  figures  we  have 

sin  (  -  0)  =  ^  =  ±1  ==  -  sin  (9, 
r        r 


cos  (  -  0 )  =  -  =    -    =       cos  0, 
r          r 


tan  (-0)=^  =  — £  =  -  tan0, 


COt    (  —  0)   =  ^—  = =r   —    COt 

y     -y 


A\       ^         r 

sec  (  —  u)  =  —  =          =       sec 

#          a; 


CSC  (  —  0)  =  —  =  — ^-  =  —  CSC  0. 

y     ~  y 

EXAMPLES. 

1.  sin  (  -  30°)  =  -  sin  30°  =  —  J, 

A/2 

2.  cos  (-45°)=      cos  45°  =  -^> 

40 

3.  tan  (-  60°)  =  -  tan  60°  =  -  V§. 

51.  Functions  of  an  angle  90°  —  6  in  terms  of  functions  of  0. 
Let  the  revolving  line  OA  (p.  76)  generate  an  angle  0,  of  any 
magnitude,  and  at  the  same  time  let  OA'  generate  an  angle 
whose  magnitude  is  90°  —  0. 

As  before,  take  OB  —  OB' ,  and  from  B,  B',  draw  perpendic- 
ulars BO,  B'C',  to  XX'.  The  triangles  OB C,  OB'C',  are,  in 
each  of  the  four  figures,  equal  geometrically.  The  proof  should 
be  supplied  by  the  student. 

With  the  same  notation  as  in  the  previous  figures  we  have, 
considering  only  the  actual  lengths  of  the  lines,  and  paying 
no  attention  to  positive  and  negative  signs,  r  =  >•',  y  =  or', 
x  —  y' . 


76 


PLANE   TRIGONOMETRY 


The  following  equations  then  hold  true  for  all  possible  cases: 

sin  (90°  -  0)  =  £-'  =  -  =  cos  0, 
r        r 


cos  (90°  -  0)  =  ±-  =  «  = 
r       r 


tan  (90°  -  0)  =  V-  =  -  =  cot  0, 
x1      y 


y 


sec  (90°  -  (9)  =  -  =  -  =  esc  0, 

x'      y 


esc  (90°  -  0)  =  -t  =  -  =  sec  0. 

' 


NOTE.  For  the  special  case  that  occurs  when  6  is  an  acute  angle,  these 
relations  were  established  independently  in  connection  with  the  definitions 
of  the  functions  of  an  acute  angle  of  a  right  triangle  (Art.  17,  p.  23). 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE 


77 


52.  Functions  of  an  angle  90°  +  6  in  terms  of  functions  of  6. 
Let  the  revolving  line  OA  generate  an  angle  0,  of  any  magni- 
tude, and  at  the  same  time  let  OA'  generate  an  angle  whose 
^nagnitude  is  90°  +  6. 

As  in  each  of  the  previous  cases,  take  OB  =  OB',  and  from 
B,  B',  draw  perpendiculars  BC,  B1  C',  to  XX'.  The  triangles 
OBC,  OB' C'  are,  in  each  of  the  four  figures,  equal  geometri- 
cally. The  proof  should  be  supplied  by  the  student. 

With  the  notation  used  in  the  previous  cases  we  have,  con- 
sidering only  the  actual  lengths  of  the  lines,  and  paying  no 
attention  to  positive  and  negative  signs,  r  =  r',x  =  yf,  y  =  xf. 
If  positive  and  negative  signs  are  taken  into  account,  these  equa- 
tions become  r  =  r',  x  =  y',  y  =  —  x'. 


The  following  equations  then  hold  true  for  all  possible  cases: 

sin  (90°  +  6)  =  '4  =    -   =      cos  0, 
r'         r 

cos  (90°  +  0)  =  ^  =  ^  =  -  sin  0, 
r        r 


78 


PLANE   TRIGONOMETRY 


tan  (90°  +  0)  =      =  —  =  -  cot  6, 
x'      -y 


cot  (90°  +  0)  =  -  =  ^  =  -  tan  6, 

' 


sec  (90°  +  9)  =  r-t  =  —  =  -  esc  0, 
x        -y 

esc  (90°  +  0)  =  -=    -    =      sec0. 

y       x 

EXAMPLES. 

1.  sin  (9(T  4-  30°)    =      cos  3CP    = 

2.  cos  (90°  +  45°)    =  -sin45°    =  -| 

3.  tan  (90°  +  60°)    =  -  cot  60°    =- 

4.  cot  (90°  +  120°)  =  -  tan  120°  =  -  (-  V3)  =  V3, 

5.  sec  (90°  +  135°)  =  ~  esc  135°  =  -  V2, 

6.  esc  (90°  +  150°)  =      sec  150°  =  -  f  V3. 

53.  Functions  of  an  angle  180°  —  9  in  terms  of  functions  of  9. 
Let  the  revolving  line  OA  generate  an  angle  0,  of  any  magni- 
tude, and  at  the  same  time  let  OA'  generate  an  angle  whose 
magnitude  is  180°  -  0. 


TRIGONOMETRIC   FUNCTIONS   OF   ANY  ANGLE  79 

As  in  the  previous  cases,  take  OB  =  OB',  and  from  B,  B', 
draw  perpendiculars  BO,  B'C',  to  XX'.  .The  triangles  OBC, 
OB'C',  are,  in  each  of  the  four  figures,  equal  geometrically. 
The  student  should  supply  the  proof. 

With  the  notation  used  in  the  previous  cases  we  have,  con- 
sidering only  the  actual  lengths  of  the  lines,  and  paying  no 
attention  to  positive  arid  negative  signs,  r  =  r',  x  ==  x',  y  =  y' . 
If  positive  and  negative  signs  are  taken  into  account,  the  second 
equation  becomes  x  =  —  x' . 

The  following  equations  then  hold  true  for  all  possible  cases  : 

sin  (180°  -  0)  =  ^  =    2    =      sin  0, 
r'         r 


cos  (180°  -  0)  =  —t  =  — -  =  -  cos  6, 
r'        r 


tan  (180°  -  0)  =  tf-  =  -£-  =  -  tan  0, 

x'       —  x 


cot  (180°  -6)  =  -  =  — -  =  -  cot  0, 

y      y 


sec  (180°  -  0)  =  r ,  =  -  -  -  -  sec  0, 
x'      —  x 


csc  (180°  -  (9)  =  -t  =    -    =      esc  0. 

1 1  y     y 

EXAMPLES. 

1.  sin  (180°  -  80°)    =      sin  30°    =      \. 

2.  cos  (180°  -  60°)    =  -  cos  60°         -  -|> 

3.  tan  (180°  -  45°)    =  -  tan  45°          -  1, 

4.  cot  (180°  -  120°)  =  -  cot  120°  =  -  f--^2)=4 

\        o   /       o 

5.  sec  (180°-  135°)  =  -  sec  135°  =  -  (- V2)=  V2, 

6.  esc  (180°  -  150°)  =      esc  150°  =      2. 


80 


PLANE    TRIGONOMETRY 


54.  Functions  of  an  angle  180°  -f  0  in  terms  of  functions  of  6. 
Let  the  revolving  line  OA  generate  an  angle  #,  of  any  magni- 
tude, and  at  the  same  time  let  OA'  generate  an  angle  whose 
magnitude  is  180°  +  6. 

As  in  the  cases  already  considered,  take  OB  =  OB',  and  from 
B,  B',  draw  perpendiculars  BC,  B'C',  to  XX'.  The  triangles 
OB 0,  OB' C' ,  are,  in  each  of  the  four  figures,  equal  geomet- 
rically. The  student  should  supply  the  proof. 


X- 


With  the  notation  used  in  the  previous  cases  we  have,  con- 
sidering only  the  actual  lengths  of  the  lines,  and  paying  no 
attention  to  positive  and  negative  signs,  r  =  rr,  x  =  x\  y  =  y*. 
If  positive  and  negative  signs  are  taken  into  account,  the  last 
two  equations  become  x  =  —  a;',  y  =  —  y*  respectively. 

The  following  equations  then  hold  true  for  all  possible  cases : 

sin  (180°  +  0)  =  tf-  =  ^  =  -  sin  <9, 
r         r 

cos  (180°  +  0)  =  ~  =  —  =  -  cos  0, 


TRIGONOMETRIC   FUNCTIONS   OF   ANY   ANGLE  81 


tan  (180°  +  0)  =    -  =         =     tan  0, 
x'       —x 

cot  (180°  +  0)  =  ^  =  —  =     cot  0, 

y'    -y 

sec  (180°  +  0)  =  ^  =  ^~  =  -  sec  0, 
x'       -y 

esc  (180°  +  0)  =  r-t  =  -±-  =  -  esc  0. 

y       -* 
EXAMPLES. 

1.  sin  (180°  +30°)   =  -  sin    30°  =  -l 

2.  cos  (180°  +  45°)   =  -cos    45°=-iV2, 

3.  tan  (180°  +60°)    =      tan    60°=      V3, 

4.  cot  (180°  +  120°)=     cotl20°  =  -iV3, 

5.  sec  (180°  +  1  35°)  =  -  sec  135°  =  -  (  -  V2)  =  V2, 

6.  esc  (180°  +  150°)=  -esc  150°  =  -2. 

55.  In  a  manner  precisely  similar  to  that  employed  in  the 
preceding  sections,  we  can  determine  the  functions  of  an  angle 
270°  —  6   in   terms  of  functions  of   6.     The  figures  for  each 
quadrant  should  be  constructed  by  the  student,  and  the  values 
obtained,  as  in  the  cases  which  have  just  been  considered. 

These  relations,  true  for  all  values  of  0,  are  as  follows  : 

sin  (270°  -0)  =  -  cos  0, 
cos  (270°  -0)=-  sin  0, 
tan  (270°  -(9)=  cot0, 
cot  (270°  -6)=  tan0, 
sec  (270°  -  0)  =  -  esc  0, 
esc  (270°  -0)=-  sec  0. 

56.  The    corresponding    values   of    functions   of   an    angle 
270°  -f  0  in  terms  of  functions  of  0  can  also  be  obtained  in  a 
manner  similar  to  that  employed  in  the  cases  already  discussed 
(Art.  50-54).     These  values  are  as  follows: 

sin  (270°  +  0)  =  -  cos  0, 
cos  (270°  +0)=  sin0, 

CONANT'S  TRIG.  —  6 


.  vV 


82  PLANE   TRIGONOMETRY 

tan (270°  +  #)  =  -cot  (9, 
cot  (270°  4-  0)=-  tan  6, 
sec  (270° +  #)=  csc0, 

esc  (270° +  0)=  -sec  (9. 
EXAMPLES. 

sin  (270°  -  210°)  =  -  cos  210°  =  -  (  -  \ V3)  =  J  V3, 
cos (270°  -  150°):=  -  sin  150°=  -  |, 
tan(270°  +  185°)  =  -cot  135°  =-(-l)=l, 
cot  (270° -2 10°  =      tan240°  =  V3» 
sec  (270° +  30°)    =      esc    30°  =  2, 
esc  (270°  +  60°)   =  -  sec    60°  =  -  2. 

57.  Functions  of  an  angle  360°  +  9  in  terms  of  functions  of  6. 

When  the  revolving  line  lias  generated  an  angle  360° +  #,  its 
position  is  the  same  as  that  occupied  after  it  has  generated  the 
angle  6.  Hence, 

The  functions  of  an  angle  360°  +  6  are  the  same  as  the  cor- 
responding functions  of  6. 

Also,  since  the  revolving  line  returns  to  its  initial  position 
after  any  number  of  complete  revolutions,  in  either  a  positive 
or  negative  direction,  it  follows  that,  when  n  is  any  positive  or 
negative  integer  or  zero, 

Functions  of  an  angle  n  x  360°  4-  6  are  equal  to  the  correspond- 
ing functions  of  6. 

In  a  similar  manner  it  may  be  shown  that  the  functions  of 
n  x  360°  —  9  are  equal  to  the  corresponding  functions  of  —  6. 

58.  By  means  of  the   equations   contained   in   Arts.   50-57, 
pp.  74-82,  the  functions  of  any  angle  can  be  found  in  terms 
of  functions  of  an  angle  less  than  90°. 

For  example,         gin  2m>  =  ^  (-  x  m,  +  3510) 

=  sin  :JJ310 
=  sin  (270°  +  81°) 
=  -  cos  81°. 
Similarly,  cos  (  -  2058°)  =  cos  2008° 

=  cos  (5  x  860°  +  258°) 
=  cos  -_>:>s 
=  cos  (270°  -  12°) 
=  -  sin  12°. 


TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE 


80 
o 


By  reductions  of  this  kind  it  is  easy  to  find  the  values  of  func- 
tions of  any  large  angle,  either  positive  or  negative.  Multiples 
of  360°  should  first  be  subtracted,  and  the  remainder  of  the 
reduction  performed  by  the  theorems  of  this  chapter. 

59.  The  following  table  contains  the  values  of  the  functions 
of  the  angles  between  0°  and  360°  which  are  of  most  frequent 
occurrence  in  elementary  mathematics. 


sine 
cosine 

tangent 
cotangent 
secant 
cosecant 


0° 

30° 

45° 

GO0 

90° 

120° 

135° 

150° 

180° 

270° 

0 

1 
2 

M 

ivs 

1 

H 

H 

1 
2 

0 

-  1 

1 

H 

H 

1 

2 

0 

i 

2 

-I* 

-H 

-1 

() 

0 

|V3 

i 

V3 

±00 

-  >/3 

-i 

> 

0 

±  CO 

±00 

V3 

i 

H 

0 

-H 

-  1 

-Va 

±   *> 

0 

1 

N 

V2 

2 

±co 

-2 

-V2 

o 

-5  VI 

-   1 

±co 

±00 

2 

V2 

h 

1 

fvs 

V2 

2 

±   CO 

_  1 

NOTE.  In  the  above  table  the  double  sign,  which  is  used  wherever  the 
value  co  occurs,  signifies  that  either  the  positive  or  the  negative  value  is 
obtained  according  as  the  revolving  line  approaches  the  given  position  from 
the  one  or  from  the  other  side.  For  example,  tan  90°  =  +  co  if  the  revolving 
line  approaches  the  positive  portion  of  the  y-axis  from  the  right,  i.e.  through 
positive  rotation;  tan 90°  =  —  co  if  the  revolving  line  approaches  the  same 
position  from  the  left,  i.e.  through  negative  rotation. 

EXERCISE  XI 
Prove  that 

1.  sin  210°  tan  225°  +  cos  300° cot  315°  =  -  1. 

2.  cos  240°  cos  120°  -  sin  120°  cos  150°  =  1. 

3.  tan  120°  cot  150°  +  sec  120°  esc  150°  =  -  1. 

4.  tan  675°  sec  540°  +  cot  495°  esc  450°  =  0. 


84  PLANE   TRIGONOMETRY 

5.  For  what  values  of  A  between  0°  and  360°  are  sin  A  and 
cos  .4  equal?     For  what  values  are  tan  A  and  cot  A  equal? 

6.  What  sign  has  sin  A  -{-cos  A  for  the  following  angles? 
A  =  120°;  A=1350;  A  =  150°;   A  =  300°;  A=315°;  A  =  690°  ; 


x      7.    What  sign  has  sin  A  —  cos  A  for  each  of  the  following 
angles?     A  =  210°;  A  =225°;  A  =  240°;  A  =  300°;  A  =625°; 

A  =  —  . 

o 

8.  What  sign  has  tan  A  —  cot  A  for  each  of  the  following 
angles?     A  =60°;  A  =  120°;    A  =135°;  A  =  150°;  A  =  210°; 

A  =  225°;  A  =  il^. 

6 

9.  Find  all  the  angles  less  than  360°  that  satisfy  the  fol- 
lowing relations  : 

=_-^?;        (6)  cos  6  =  -  ^?;        <V)  tan  6  =  -  1  ; 
V3.     2 

10.    Prove  sec  (A  -  180°)  =  -  sec  A. 
—  -  11.    Prove  cot  (A  -  270°)  =  -  tan  A. 

12.  Prove  cos  A  +  cos  (90°  +  A)  +  cos  (180°  -  A) 

-cos  (270°-  A)=0. 

13.  Prove 

----  ' 

tan  (180°  +  A}      cot  (270°  -  A)     sec  (360°  -A)  A 

tan  (180°  -A)  '  cot  (270°  +  A)  '  esc  (360°  +  A) 

14.  Find  the  value  of 

sin(—  A)          cos  (  —  A)          tan(—  A) 
cos(90°  +  A)      sin  (90°  +  A)      cot  (90°  +  A)  ' 

15.  Express  in  simplest  form 

cos  (180°  -  A)  §  tan  (270°  -  A)  t  gec^ 
sin  (180°  +  A)   '  cot  (270°  +  A)  ' 


\ 


CHAPTER   VII 

GENERAL    EXPRESSION    FOR    ALL    ANGLES    HAVING    A 
GIVEN   TRIGONOMETRIC    FUNCTION 

60.  From  the  definitions  of  the  trigonometric  functions  it  is 
evident  that  a  given  angle  can  have  but  one  sine,  one  cosine, 
one  tangent,  etc. 

But  the  converse  statement  is  not  true.  A  given  sine  may 
belong  to  any  one  of  an  infinite  number  of  angles.  The  same 
is  true  of  the  cosine,  the  tangent,  or  of  any  of  the  other  trigo- 
nometric functions.  This  has  already  been  alluded  to  inci- 
dentally (Arts.  50-57,  pp.  74-82).  Expressions  will  now  be 
found  for  all  angles  that  have  a  given  sine,  a  given  cosine,  a 

given  tangent,  etc. 

i 

61.  When  the  revolving  line  has  made  one  complete  revolu- 
tion in  either  direction,  it   has  generated   an  angle   of   ±  2  TT 
radians ;  when  it   has  made  two  complete  revolutions,  it  has 
generated  an  angle  of  ±  4  TT  radians ;  and,  in  general,  when  it 
has  made  three,  four,  five,  etc.,  revolutions,  it  has  generated 
an  angle  of  ±  6  TT,  ±  8  TT,  ±  10  TT,  etc.,  radians. 

These  statements  may  be  combined  into  a  single  expression 
by  means  of  the  following  statement : 

When  the  revolving  line  has  made  any  number  of  complete 
revolutions  in  either  direction,  it  has  generated  an  angle  of  2  nir 
radians,  where  n  is  some  positive  or  negative  integer  or  zero. 

62.  General  expression  for  all  angles  that  have  the  same  sine. 
Let  XOA  (p.  86)  be  any  convenient  angle,  a,  and  let  XOA! 
be  equal  to  TT  —  a.     By  Art.  53,  the  sine  of  XOA  =  the  sine  of 
XOA' ;   or  sin  a  =  sin  (TT  —  a).      Also,  TT  —  a  is  the  only  other 
angle  between  0°  and  360°,  or  between  0  and  2  TT  whose  sine  is 
equal  to  the  sine  of  a.      But  (Art.  61)  any  angle  whose  initial 
line  coincides  with  OX  and  whose  terminal  line  also  coincides 

85 


86 


PLANE   TRIGONOMETRY 


with  OX  is  represented  by  the  expression  2  mr.  Hence,  all 
angles  whose  initial  lines  coincide  with  OX  and  whose  terminal 
lines  coincide  with  OA  are  represented  by  the  expression 
2  ntr  +  «. 

Any  angle  whose  initial  line  coincides  with  OX  and  whose 
terminal  line  coincides  with  OX'  is  represented  by  the  expres- 
sion 2n7r-h  TT,  or  (2w  4-  1)  TT.  Hence,  all  angles  whose  initial 
lines  coincide  with  OX  and  whose  terminal  lines  coincide  with 
OA'  are  represented  by  the  expression  (2w  +  I)TT—  a.  These 
two  expressions,  2  mr  +  a  and  (2  n+  I)TT  —  «,  are  both  included 
in  the  more  general  expression  ynr  -f(  — l)n«;  that  is,  a  is  to  be 
added  to  any  even  multiple  of  TT,  and  subtracted  from  any  odd 
multiple  of  TT.  This  will  be  understood  if  successive  values 
are  substituted  for  71,  and  the  resulting  positions  of  the  ter- 
minal line  noted.  This  is  conveniently  done  by  means  of  the 
following  table  : 

If  w  =  0,  mr  +(—!)"«=  «, 

n  =  1,  UTT  4-  (  —  1 )"«  =     TT  —  a, 

71  =  2,  W-7T  +  (  —  1  )7i«  =  2  7T  +  0, 

n  =  3,  .WTT  +  (  -  !)"«  =  3  TT  -  a, 

71  =  4,  WTT-f-  (  —  !)"«  =  4-7T  -|-  «, 

n  =  5,  7i?r  4-  (  —  l)na  =  5  TT  —  a, 

71  =  6,  TiTT  -f  (  —  1  )"«  =  6  7T  +  «, 

n  =  l,  UTT  -f  (  —  1 )"«  =  7  TT  —  a, 

W=8,  /Z7T  +(-!)»«=  8  7T+a, 

7i  =  9,  mr  +  (  —  1  )na  =  9  TT  —  a, 


71=  -1, 

71=  -2, 


H7T  +  (—  l)n«  =   —  7T  —  «, 

rc-TT  +  (  -  l)na  =  -  2  TT  +  a. 

In  this  table  we  observe  that 
whenever  n  is  an  even  number,  the 
expression  (—  l)n=-f-l,  and  the 
angle  that  the  revolving  line  has 
then  generated  is  (Art.  61,  p.  85)  a 
certain  number  of  complete  revolu- 
tions plus  the  angle  «.  If  n  is  an  odd 
number,  the  expression  (  —  1)"=  —1, 
and  the  angle  that  the  revolving 


GENERAL   EXPRESSION   FOR  ALL   ANGLES  87 

line  has  generated  is  a  certain  number  of  complete  revolutions 
plus  a  half  revolution,  minus  the  angle  a.     That  is, 

The  expression  n-rr  +(—!)"<*  is  a  general  expression  for  all 
-angles  that  have  the  same  sine  as  the  angle  a. 

63.  In  this  connection  it  should  be  noted  that,  when  n  is 
any  positive  or  negative  integer  or  zero,  2  n  is,  by  definition,  an 
even  number,  and  2  n  -f-  1  is  an  odd  number. 

64.  General  expression  for  all  angles  that  have  the  same  cosine. 

The  cosine  of  the  angle  360°  —  a,  or  2  TT  —  «,  is  equal  to  the 
cosine  of  the  angle  a ;   and  2  TT  —  a 
is  the  only  angle  between  0  and  2  TT 
that    has    the    same    cosine    as    the 
angle  a. 

But,  reasoning  in  the  same  manner  X- 
as  in  Art.  62,  all  angles  whose  initial 
lines  coincide  with  OX  and  whose 
terminal  lines  coincide  with  OA  are 
included  in  the  expression  2  mr  +  a ; 
and  all  angles  whose  initial  lines  coincide  with  OX  and  whose 
terminal  lines  coincide  with  OA'  are  included  in  the  expression 
2  mr  —  a.  Hence, 

The  expression  2  mr  ±  a  is  a  general  expression  for  all  angles 
that  have  the  same  cosine  as  the  angle  a. 

65.  General   expression  for  all  angles  that  have    the   same 
tangent.     The  tangent  of  180°  +  «,  or  TT  -f-  «,   is  equal  to  the 
tangent  of  a ;  and  TT  +  a  is  the  only  angle  between  0  and  2  TT 
that  has  the  same  tangent  as  the  angle  a  (see  fig.  p.  88). 

All  angles  whose  initial  lines  coincide  with  OX  and  whose 
terminal  lines  coincide  with  OA  are  included  in  the  general  ex- 
pression 2  mr  +  a,  and  all  angles  whose  initial  lines  coincide 
with  OX  and  whose  terminal  lines  coincide  with  OA'  are  in- 
cluded in  the  general  expression  (2  n  +  I)T  +  <*•  But,  since  2  n 
signifies  only  even  integers,  and  2  w  -f  1  only  odd  integers,  while 
n  includes  all  integers,  both  even  and  odd,  the  two  expressions, 
2  HTT  +  a  and  ('2  n  +  I)TT  -|-  «,  can  be  combined  as  follows  : 

The  expression  HTT  -{-  a  is  a  general  expression  for  all  angles 
that  have  the  same  tangent  as  the  angle  a. 


88 


PLANE   TRIGONOMETRY 


66.  Since  cot  «  is  the  reciprocal  of  tan  a,  the  general  expres- 
sion for  all  angles  that  have  the  same  cotangent  as  .the  angle  a 

is  HTT  +  a. 

Since  sec  a  is  the  reciprocal  of 
cos  a,  the  general  expression  for  all 
angles  that  have  the  same  secant  as 
-X  the  angle  a  is  2n7r±a. 

Since  esc  a  is  the  reciprocal  of 
sin  «,  the  general  expression  for  all 
angles  that  have  the  same  cosecant 
as  the  angle  a  is  mr  +  (  —  l)na. 

67.  In  the  following  examples,  and  in  practical  work  generally, 
the  smallest  positive  value  of  a  is  taken.     This  is  done  simply 
for    convenience,    the   results   just   obtained    being    perfectly 
general. 

EXERCISE  XII 

1.  What  is  the  general  expression  for  all  angles  whose  sine 
is  J? 

The  smallest  positive  angle  whose  sine  equals  £  is  30°,  or  -. 

6 

.-.  0  =  —  is  the  smallest  positive  angle, 

and   (Art.   62)   0=  WTT+(  —  1)"—  is  the  general  expression  for  all  angles 
whose  sine  is  }. 

2.  What   is    the   general   expression   for   all   angles  whose 
tangent  is  V3  ? 

The  smallest  positive  angle  whose  tangent  is  x/3  is  60°,  or  ^ . 

3 

.-.  0  —  ^  is  the  smallest  positive  angle, 
o 

and    (Art.  65)   6  =  mr  +  ^  is  the  general  expression  for  all  angles  whose 
tangent  is  V3. 

3.  What  is  the  general  expression  for  all  angles  whose  cosine 
is  —  J,  and  whose  tangent  is  —  V3? 

The  only  angles  between  0°  and  360°  whose  cosine  is  -\  are  120°  and  240°. 
The  only  angles  between  0°  and  360°  whose  tangent  is  -  V3  are  120°  and 
300  . 

The  only  angle  that  satisfies  both  these  conditions  is  120°,  or  |  TT. 

.-.  0=  2n>rr+  ITT. 
Another  general  expression  for  the  same  angles  is  (2  n  -f  1)  TT  —  |  TT. 


GENERAL   EXPRESSION    FOR  ALL   ANGLES  89 

Find  the  general  value  of  0  which  satisfies  each  of  the  follow- 
ing equations  : 

4.  sin0=lV2.  13.  tan0  =  —  JV3. 

5.  sin  0=1.  14.  sin20  =  £. 

6.  sin0=-|V8.  15    COS20  =  |. 

7'  sin*  =  -J-  16.  8  tan2  0  =  1. 

8.  COS0=W3. 

17.  3  sec2  0  =  4. 

9.  cos0  =  -1|V2. 

18.  cot2  0  =  1. 

10.  cos  0  =  0. 

11.  cos0=-l.  19-  tan2  0=2  sin2  0. 

12.  tan  0  =  1.  20.  2  tan2  0  =  sec2  0. 

21.  What  is  the  general  value  of  0  that  satisfies  both  of  the 
following  equations  ? 

sin  0  =  J  V3,  and  cos  0  =  J. 

22.  What  is  the  general  value  of  0  that  satisfies  both  of  the 
following  equations? 

sin  0  =  —  l,  and  cos  0  =  —  J  V3. 

In  the  following  five  examples,  show  that  the  same  angles 
are  indicated  by  both  the  given  expressions. 

23.    mr  +  -,  and 


24.  W7r+(_ 

25.  M7T  —  ^,   and  WTT  +  TT 

6  6 


26.    WTT  +  ^,  and  —  mr  + 
o  o 


27. 


\\ 


90  PLANE   TRIGONOMETRY 

68.  An  equation  involving  trigonometric  functions  of  an 
unknown  angle  is  called  a  trigonometric  equation. 

The  solution  of  a  trigonometric  equation  involves  the  de- 
termination of  all  angles  that  satisfy  the  equation. 

In  solving  a  trigonometric  equation,  the  smallest  positive 
angle  that  satisfies  it  should  first  be  determined,  and  then  the 
general  value  should  be  found  for  all  angles  that  satisfy  it. 
This  has  been  illustrated  in  the  examples  of  Exercise  XII,  and 
will  be  still  further  shown  in  those  of  the  following  set. 

EXERCISE   XIII 

1.    Solve  the  equation  cos2  6  +  2  sin2  6  =  J. 

This  may  be  written 

2-2cos20  =  £. 
.-.  cos'2  0  =  f  . 


The  smallest  positive  angle  whose  cosine  is  -  V3  is  30°,  or  -. 

Therefore,  using  the  positive  result,  0  =  2  rnr  ±  - 

Al§p,  the  smallest  positive  angle  whose  cosine  is  -  £\/3  is  150°,  or  f  TT. 

Therefore,  using  the  negative  result,  0  =  2  rnr  ±  |  TT,  or  (2  n  +  1)  TT  ±  -. 

6 
These  two   sets   of   values   may  be  combined  in  the  single  expression 


2.    Solve  the  equation  2  cos2  0  —  V3  sin  0  +  1  =  0. 
This  may  be  written 

2-2  sin2  0  -  V3  sin  6  +  1  =  0. 

2  sin2  0  +  V3  sin  0-3  =  0. 

Factoring,  (sin  0  +  V3)  (2  sin  0  -  V3)  =  0. 

.-.  sin  0  =  -  V3,  or  sin  0  =  |  V3. 

The  sine  of  an  angle  cannot  be  numerically  greater  than  1;  therefore,  the 
first  equation  gives  no  solution. 

The  smallest  positive  angle  that  satisfies  the  equation 
sin  0  =  \  V3, 

is  0  =  60°,  or  5, 

o 

and  (Art.  62)  the  general  expression  for  the  value  of  all  angles  that  have  the 
same  sine  as  60°  is  TT 

8  =  n7r+  (~l)n|- 

Therefore,  the   most  general   expression  for  all  angles  that  satisfy  the 
original  equation  is  mr  +  (—  l)nf  • 


GENERAL  EXPRESSION  EOR  ALL  ANGLES       91 

3.    Solve  the  equation  tan  4  0  —  cot  3  6. 

This  may  be  written  tan  4  0  =  tan  I  *  -  3  0\  by  Art.  51, 

=  tan  (nir  +  f  -  3  6\  by  Art.  65. 


.-.  4  0  =  riTT  +  -  -  3  0, 


Solve  the  following  equations,  finding  the  general  value  of 
in  each  case  : 

4.  2  sin2  0-  cos  0  =  1.  16.  sin  30  =  sin  90. 

5.  tan2  0  +  sec  0  =  1.  17.  cos  6  0  =  cos  2  0. 

6.  cot2  0-  esc  0  =  1.  18  Cos40  =  cos50. 

7.  cos2  0  -  sin  0=1. 

19.  cos  mv  =  cos  r&0. 

8.  2  sin2  0  +  3  cos  0  =  0. 

20.  cos  4  0  =  sin  2  0. 

9.  2  cos2  0  +  cos  0=  1. 

21.  sin  4  .0  =  cos  2  0.. 

10.  sin2  0  -  2  cos  0  +  I  =  0. 

11.  3sin20-2sin0=l.  22.  "tan  2  0  =  tan  30. 

12.  23>  cot  5  6  =  COt  2 


13.  csc20-cot0=3.  24-    tan  4  0  =  cot  5  0. 

14.  tan2  0  +  cot2  0  =  2.  25.    tan  ^0  4-  cot  nO  =  0. 

15.  sin  50=  sin  2  0.  26.    tan  2  0  tan  0  =  1. 


M 


CHAPTER  VIII 


RELATIONS  BETWEEN  THE  TRIGONOMETRIC  FUNCTIONS 
OF   TWO   OR   MORE   ANGLES 

69.   Sine  and  cosine  of  the  sum  of  two  angles.     Let  x  and  y 

be  acute  angles,  and  let  x  -f  y  be  either  acute  or  obtuse.  In 
both  figures  the  lettering  is  so  arranged  that  the  following 
demonstrations  apply  to  either  case, 


o 


D  F 


o  F 


=  Z 


From  C,  any  point  in  OB,  draw  CD  _L  XX',  and  CE  JL  OA  ;  and  from  E 
draw  EH  \\  XX'  and  EF±XX'. 
Since  Z  x  -  Z  OEH  =  90°  - 

..-.  Z.x  =  Z.HCE. 

T\f~V 

Then  we  have  sin  (x  +  y)  =  =-:- 

^        J)      OC 


Also, 


OC 

=  oc  +  oc 

~  ~OE  ~OC  +  ~CE  OC 
=  sin  x  cosy  +  cosZ  HCE  sin  y. 
.  .  sin  (a?  +  ?/)  =  sin  x  cos  y  +  cos  a?  sin  y. 

OP 

OC 

OF  -  DF 


(1) 


cos  (x  +  y}  = 


OC 

=  OF      HE 
OC      OC 

=  OF  OE      HE  CE 
OE  OC      CE  OC 
—  cos  x  cos  y  —  sin  Z  HCE  sin  y. 
cos  (x  +  y)  =  cos  as  cos  y  —  sin  a?  sin  ?/. 
92 


(2) 


RELATIONS   BETWEEN   TWO   OR  MORE   ANGLES         93 

70.  The  above  proofs  are  given  only  for  the  case  when  both  x 
and  y  are  acute. 

To  prove  the  formulas  true  for  all  values  of  x  and  y  we 
_proceed  as  follows : 

Let  x  and  y  be  acute  angles,  and  let  xl  =  90°  +  x;  .then  we  have  (Art.  52), 
sin  x}  =  cos  x,  and  cos  a^  =  —  sin  x*  (1) 

Then,  sin  (aj  +  #)  =  sin  (90°  +  x  +  y) 

=  cos  (z  +  ?/),  (Art.  52)     (2) 

where  #  and  y  are  both  acute  angles. 

But  (Art.  69,  p.  92)  when  #  and  y  are  both  acute  angles, 
cos  (x  +  y)  —  cos  x  cos  y  —  sin  #  sin  #. 

Substituting  in  this  equation  the  values  given  in  (1)  and  (2),  we  have 

sin  (a?i  +  y}  —  sin  a?i  cos  y  +  cos  a?i  sin  ?/.  Q.E.D. 

In  like  manner,        cos  (x^  +  ?/)  =  cos  (90°  +  a;  +  ?/) 

=  -sin(a;  +  y),  (Art.  52)     (3) 

where  x  and  y  are  both  acute  angles. 

But  (Art.  69,  p.  92)  when  x  and  y  are  both  acute  angles, 

—  sin  (a:  +  ?/)  =  —  sin  #  cos  ?/  —  cos  x  sin  y. 

Substituting  in  this  equation  the" values  given  in  (1)  and  (3),  we  have 

cos  (a?i  +  t/)  =  cos  xi  cos  ?/  —  sin  oc\  sin  ?/.  Q.E.D. 

Formulas  (1)  and  (2)  (Art.  69,  p.  92)  have  now  been 
proved  for  the  case  when  x  is  obtuse  and  y  is  acute. 

Letting  y^  =  (90°  +  «/),  and  proceeding  in  the  same  manner, 
we  can  establish  these  formulas  for  the  case  when  both  angles 
are  obtuse. 

Then,  letting  a;2=900  +  a:1,  ^  =  90°  +  ^,  xs=90°-\-x^  etc., 
and  proceeding  in  a  precisely  similar  manner,  we  can  establish 
the  formulas  for  all  possible  values  of  x  and  y. 

71.  Sine  and  cosine  of  the  difference  of  two  angles.     Let  x  and 

y  be  two  acute  angles,  placed  as  represented  in  the  figure.     It 
is  here  assumed  that  x  >  y. 

From  C,  any  point  in  the  final  position  of  the 
generating  line  OA,  draw  CD  1  OX  and  CE  A.  OB. 
Prolong  DC,  and  from  E  draw  EH  \\  OX,  inter- 
secting DC  produced  in  H. 

Since  /• 

X 


94  PLANE   TRIGONOMETRY 

Then,  sin  (x  -  y)  =  — - 

=  FE-  HC 

OC 

=  FE  OE      HC  EC 
OE  OC      EC  OC 
—  sin  x  cos  y  —  cos  Z  ECU  sin  y. 

.-.  sin  (a?  -y}=  sin  a?  cosy  —  cosa?siny.  (1) 

In  like  manner,  nn 


OF  + 


06' 

=  OF      EH 
OC      OC 

=  OFOEKH  EC_ 
OE  OC      EC  OC 
=  cos  x  cos  ij  +  sili  Z  EC/7  sin  y. 
.  :  cos  (a?  —  y  )  =  cos  .»  cos  ?/  +  sin  x  sin  ?/.  (2) 

These  proofs  have  been  given  on  the  assumption  that  x  >  y. 
To  prove  that  they  are  true  when  x  <  y,  we  proceed  as  follows  : 
sin  (x  -,  y}  =  sin  [  -  (y  -  x)] 

=  —  sin  (?/  —  #),  (Art.  50,  p.  75) 

=  —  sin  ^  cos  £  -f  cos  y  sin  x, 

or,  rearranging  the  terms  and  the  factors  in  each  term, 

sin  (a?  —  y)  =  sin  a?  cos  2/  —  cos  a?  sin?/.  Q.  E.  D.     (3) 

In  like  manner, 

cos  (x  -  y}  =  cos  \_-(y  -  x~}~\ 

-  cos  (y  -  x)  (Art.  50) 

=  cos  y  cos  x  -f-  sin  ?/  sin  #, 

or,  rearranging  the  factors  in  each  term, 

cos  (x  —  y}  =  cos  x  cos  y  +  sin  a?  sin  y.  Q.  E.D.     (4) 

72.  The  formulas  of  Art.  71  have  now  been  proved  for  all 
cases  when  x  and  y  are  both  acute  angles.  To  prove  that  they 
are  true  for  all  possible  values  of  x  and  ?/,  we  proceed  as 
follows  : 

Let  x  and  y  be  acute  angles,  and  let  x\  —  90°  +  x.     Then, 

sin  XL  =  cos  x,  and  cosa,^  =  —  sin  a:.  (1) 

Then  we  have    sin  (xl  —  y)  =  sin  (90°  +  x  —  y) 

=  cos(x  -  y).  (Art.  52)     (2) 


RELATIONS    BETWEEN    TWO   OR    MORE   ANGLES          95 

But  since  .r  and  //  are  acute  angles, 

cos  (x  —  y}  =  cos  x  cos  y  +  sin  x  sin  y.  (3) 

Substituting  in  (3)  the  values  given  in  (1)  and  (2),  we  have 

sin(a?i  —  y}  =  sin  a?i  cosy  -  cos^isiny.  Q.  E.  D.     (4) 

In  like  manner, 

cos  (.rt  -  y)  =  cos  (90°  +  x  -  y)  =  -  sin  (x  -  y).  (5) 

But  since  x  and  y  are  acute  angles, 

-  sin  (x  -  y}  =  -  (sin  x  cos  y  -  cos  x  sin  y). 

(Art.  71,  p.  94)     (6) 
Substituting  in  (6)  the  values  given  in  (1)  and  (5),  we  have 

cos  (a?i  —  y)  =  cos  a?i  cos  y  +  sin  a?!  sin  t/.  Q.  E.  D.     (7) 

Formulas  (1)  and  (2)  (Art.  71,  p.  94)  have  now  been  proved 
for  the  case  when  x  is  an  obtuse  angle  and  y  is  an  acute 
angle. 

Letting  y1  =  90°  +  ?/,  and  proceeding  as  before,  we  can  es- 
tablish these  formulas  for  the  case  when  both  angles  are 
obtuse. 

Then,  letting  x2  =  90°  +  ^,  #2  =  90°  +  yv  x8  =  90°  +  xv  etc., 
and  proceeding  in  a  precisely  similar  manner,  we  can  establish 
the  formulas  for  all  possible  values  of  x  arid  y. 

EXERCISE   XIV 

1.  Find  the  value  of  sin  75°. 

sin  75°  =  sin  (45°  +  30°) 

=  sin  45°  cos  30°  +  cos  45°  sin  30° 

=  _L  ^1  +  JL1 

V2    2         \/:2  2 
=  V3  +  1 

2\/2 

2.  Find  the  value  of  sin  15°. 

si nl 5°  =  sin  (45°  -30°) 

=  sin  45°  cos  30°  -  cos  45°  sin  30° 
1     \/3        1    1 

~  \/2    2        V2  2 
=  V3-1 

2V2 


96  PLANE    TRIGONOMETRY 

3.    Find  the  value  of  cos  105°. 

cos  105°  =  cos  (60°  +  45°) 

=  cos  60°  cos  45°  -  sin  60°  sin  45° 

=  i  i      Va  i 

2  V2        2     V2 


2  "s/2 

4.  If  sin  a  =  |  and  sin  /3  =  ^|,  find  sin  («  —  /3). 

5.  If  sin  a  =  |  and  cos  ft  =  if,  find  cos  (a  -f  /:?). 
^ 6.    If  cos  a  =  £•&,  and  cos  ft  =  -|,  find  cos  (a  —  /3). 

Prove  that 

7.  si 

8.  sin  105°  + cos  105°  =  cos  45°. 

9.  sin  75°-  sin  15°  =  cos  105°  +  cos  15°. 

10.  sin  (45°  -  6)  cos  (45°  -</>)-  cos  (45°  -  0)  sin  (45°  -  0) 

=  sin  ((/>  —  6). 

HINT.  Let  #  =  45°  -  0  and  y  =  45°  -  <£.  Then  compare  with  (1), 
Art.  66.  The  converse  application  of  the  x-y  formulas,  as  illustrated  by 
this  example,  is  of  frequent  occurrence. 

11.  sin  (45°  +  6)  cos  (45°  -  (£)  +  cos  (45°  +  0)  sin  (45°  -  0) 

=  cos  (6  —  $). 

13.  cos  (30°  +  «)  cos  (30°-  a)  +  sin  (30°  +  «)  sin  (30°  -  «) 

=  cos  2  a. 

14.  cos  a  cos  (/3  —  «)  —  sin  «  sin  (/3  —  a)  =  cos  /3. 

15.  sin  (n  +  1)«  sin  (w  —  1)«  +  cos  (n  +  1)<*  cos  (w-  —  1)« 

=  cos  2  «. 

16.  sin  (n  +  l)a  sin  (n  +  2)a  -f  cos  (n  +  1)«  cos  (n  +  2)« 

=  cos  a. 

17.  sin  (a  -  jB  +  15)  cos  (/3  -  «  +  15) 

-  cos  (a  —  0+  15)  sin  (/£  —  a  +  15)  =  sin  (2  «—  2  yS). 


RELATIONS   BETWEEN   TWO   OR   MORE   ANGLES          97 

The  following  examples  are  of  especial  importance,  and  are 
often  used  as  standard  formulas. 

18.  sin  75°  =  cos  15°  = 


_ 

2V2 

19.  sin  15°  -  cos  75°  - 


2V2 

20.  cos  (a?  +  y)  cos  (a?  —  y)  =  cos2  a?  —  sin2  y. 

21.  sin  (a?  +  -*/)  sin  (ac  —  y}  =  cos2  y  —  cos2  sc. 

73.  Tangent  of  the  sum  and  of  the  difference  of  two  angles. 

For  all  values  of  x  and  y  we  have  (Art.  69) 

sin  (x  +  y)  =  sin  a;  cos  y  +  cos  a:  sin  y, 
and  cos  (a;  +  y)  =  cos  #  cos  y  —  sin  a;  sin  y. 

tan  Q  +  y)  ^  sin  *'  cos  ^  +  cos  x  sin  ^  . 
cos  x  cos  y  —  sin  x  sin  ?/ 

Dividing  both  numerator  and  denominator  by  cos  a:  cosy,  we  have 

sin  x  cos  y      cos  ar  sin  y 

,.      cos  a:  cos  \i      cos  a:  cos  y 
tan  (a?  -f  y)  = 


1  —  tan  a?  tan  y 

In  like  manner,  . 


sn  arsn 
cos  a:  cosy 

(1) 


tan    *  - 


cos  (a:  -  y) 

_  sin  x  cos  y  —  cos  x  sin  y 
cos  x  cos  y  +  sin  x  siti  y 

sin  a;  cos  y  cos  a:  sin  y 

_  cos  a;  cos  y  cos  a:  cos  y 

~~  cos  a:  cos  y  sin  a:  sin  y 

cos  a;  cosy  cos  a:  cosy 

sin  a:  _  sin  y 
cos  a:      cos  y 


. (2) 
1  -f  tan  35  tan  i/ 


CONANT  S    TRIG.  — 


98  PLANE    TRIGONOMETRY 

74.   Cotangent  of  the  sum  and  of  the  difference  of  two  angles. 

For  all  values  of  x  and  y  we  have 


Expanding    cos  (x  -f  ?/)     and     sin  (x  -f-  ^),     dividing     both 
numerator  and  denominator  by  sin  x  sin  ?/,  and  reducing,  we 


-  (1) 

coty 

In  a  similar  manner  it  can  be  proved  that 


uin*^  \j\rii  y  ~t  /"O\ 

coty  —  cota? 

75.  Formulas  (1)  and  (2),  Art.  69,  (1)  and  (2),  Art.  71, 
(1)  and  (2),  Art.   73,  and  (1)  and  (2),  Art.  74,  are  often  re- 
ferred to  as  the  addition  and  subtraction  formulas.     The  addi- 
tion formulas  are  sometimes  known  as  the  x  +  y  formulas,  and 
the  subtraction  formulas  as  the  x  —  y  formulas.     When  refer- 
ence is  made  to  both  groups  together,  the  general  expression, 
"the  x-y  formulas,"  is  often  employed. 

76.  From  the  formulas  for  the  functions  of  the  sum  of  two 
angles  the  formulas  for  the  functions  of  the  sum  of  three  angles 
are  at  once  obtained,  as  follows : 

sin  ( x  +  y  +  z)  =  sin  [(x  +  y)  +  z] 

—  sin\(.r  +  y)  cos  z  -f  cos  (x  +  /y)  sin  z\ 
=  (sin  x  cos  y  +  cos  x  sin  y)  cos  z 

+  (cos  x  cos//  —  sin  a:  sin  y}  sin  z. 
.*.  sin  (x  +  y  +  z)  =  sin  a;  cos//  cos  z  -f  cosx  siny  cosz 

-f  cos  x  cosy  sin  z  —  sin  x  siuy  sinz.  (1) 

In  like  manner  it  can  be  proved  that 

cos  (x  +  y  +  z)  —  cos  x  cosy  cosz  -  cosx  siny  sinz 

—  sin  x  cosy  sin  z  —  sin  x  sin  y  cosz,  (2) 
and  that 

cos  (x  +  y  +  z) 

_   tan  x  +  tan  y  +  tan  z  —  tan  x  tan  /y  tan  z          ,.,, 
1  —  tan  x  tan  y  -  tan  x  tan  z  —  tan  y  tanz 


^ 


RELATIONS   BETWEEN   TWO   OK  MORE   ANGLES         99 

EXERCISE   XV 

1.  If  tan  «  =  |  and  tan  ft  —  1,  iind  tan  («  +  ft). 

2.  If  tan  a  =  |  and  tan  /3  =  f  f,  find  tan  (£_«). 

3.  If  tan  «  =  f  and  cot  ft  =  -f%,  find  cot  (a  -f  /8). 

4.  If  tan  «  =  |  and  ft  =  45°,  find  tan  («  +  /3). 

5.  If  tan  «  =  -|-  and  tan  /8  =  £,  find  tan  (2  a  -f  ft). 

6.  If  tan  a  =  —^—  and  tan  /9  =  -  -  -  ,  find  tan  (a  +  ft). 

n  +  l  2  W  -f  1 

7.  If  tan  a  =  |  and  tan  ft  =  Jj,  prove  that  a  -f  ft  —  45°. 


The  next  four  examples  are  of  especial  importance,  and  are 

«    tan  75"  =  cot  16«  =  2 


often  used  as  standard  formulas. 


8. 

,        ,  cot  (9-1 

12.    cot  -  +      = 


10.    tanl5°-cot75°=2-V3. 


14.    tan [ —  -f 

15.  Prove  the  identity  cos  («  4-  ft)  cos  ft  -+-  sin  (a  +  ft)  sin  ft  = 
cos  a. 

HINT.  Let  a  +  /?  =  x  and  /?  =  y.  Then  compare  with  (2),  Art.  69. 
Many  of  the  remaining  examples  can  be  worked  without  difficulty  by 
applying  the  addition  or  subtraction  formulas  directly. 

16.  sin  2  a  cos  a  -f-  cos  2  a  sin  a  =  sin  3  a. 

17.  sin  3  a  cos  «  —  cos  3  a  sin  «  =  sin  2  a. 

18.  cos  3  a.  cos  2  a  —  sin  3  a  sin  2  a  =  cos  5  a. 


sec  «         esc  (x. 

20.  sin  (60°  +  «)  cos  (30°+  a)  -  cos  (60°  +  a)  sin  (30°  +  a)  =  J. 

21.  t»in2«+tan«   =  tftn  3  g> 
1  —  tan  2  «  tan  a 


22.  -         =tan2c. 

1  —  Un(«  +  /8)tan(«  -  ft) 


100  PLANE    TRIGONOMETRY 

tana-  t»u    .- 


23. 


1  +  tan  a  tan  (a  — 


cot  3  a  cot  2  a  4-  1 

24.  —  =  cot  a. 
cot  2  a  —  cot  3  a 

25.  tan  2  0  -  tan  6  =  tan  0  sec  2  0. 

26.  sec  2  6  =  1  +  tan  2  0  tan  0. 

27.  csc20  =  cot0-cot20. 

tan  3  6  —  tan  2  0        tan  4  6  —  tan  3  0 


tan  3  0  tan  20      1  +  tan  4  0  tan  3  0 

4tan0 


29.    tan  (45°  4-  0)  -  tan  (45°  -  0)  = 


JL  —  tan  u 
sin  (#  4-  y)        cot  #  4-  cot  y 

3O.      -  ^  -  —  —  =  -  —  —  • 

cos  (x  T-  y)      1  4-  cot  #  cot  y 

77.   The  algebraic  sum  of  two  sines  or  of  two  cosines  in  the 
form  of  a  product.     For  all  values  of  x  and  y  we  have  (Arts.  69 

sin  (x  4  ?/)  =  sin  x  cosy  4  cos  x  sin  y, 
and  sin  (a;  —  y)  =  sin  a:  cos  y  —  cos  #  sin  y. 

Adding  and  subtracting,  we  have 

sin  (x  4  y)  4  sin  (a;  —  y}  =  2  sin  #  cos  y,  (1) 

and  sin  (x  4  #)  —  sin  (x  —  #)  =  2  cos  x  sin  y.  (2) 

Also  (Arts.  69  and  71), 

cos  (a:  4  y)  =  cos  x  cosy  —  sin  a;  sin  y, 
and  cos  (a?  —  y)  =  cos  a:  cos  y  4  sin  re  siny. 

Adding  and  subtracting,  as  before,  we  have 

cos  (x  4  y)  4  cos  (ar  —  ?/)  =  2  cos  a:  cos  y,  (3) 

and  cos  (a;  4  y)  —  cos  (a;  —  y~)  =  —  2  sin  a;  sin  y.  (4) 

Let  a:  4-  y  —  u,  and  x  —  y  =  v. 

Solving  these  two  equations  for  x  and  y, 


. 
Substituting  these  values  of  a:  and  ?/  in  (1),  (2),  (3),  and  (4),  we  have 

smu  4  sinv  =  2sm^^cos^^$  (5) 

**• 

A^--\r 

sin  w  -  sin  v  =  2  cos  ^  +  ^  sin^-^T  (6) 

cost*  +  cost?  -  2cos^pcos^p;  (7) 

cost*  -  cos  v  =  -  2sin^±^  sin-^.  (8) 


RELATIONS   BETWEEN   TWO   OR   MORE   ANGLES      ;I01 

These  formulas  are  among  the  most  important  of  all'  the 
formulas  of  trigonometry.  The  student  should  commit  them 
carefully  to  memory,  and  become  perfectly  familiar  with  their 
-application.  They  will  sometimes  be  referred  to  as  the  u-v 
formulas. 

As  illustrations  of  the  manner  in  which  certain  expressions 
can  be  simplified  by  the  application  of  one  or  more  of  these 
processes,  the  following  examples  are  given  : 


2. 


sin  75°  -  sin  1 5°  _ 


2  2 

=  2  cos  40°  sin  30° 
=  cos  40°. 

75°  +  15°   .    75°  -15° 
2  cos sin 


cos  75°  + cos  15°     n_  _75°  +  15°        75°  -  15° 

•  cos _ 


2 

_  2  cos  45°  sin  30° 
~  2  cos  45°  cos  30° 

=  tan  30° 

=  iV3=  0.57735. 


(sin  6  0  +  sin  2  0)(cos  2  0  -  cos  4  6) 
(sin  5  6  +  sin  0)(cos  3  0  -  cos  5  0) 

_  (2  sin  4  0  cos  2  0)(2  sin  3  0  sin  0)  = 
~  (2  sin  3  0  cos  2  0)  (2  sin  4  9  sin  0) 

EXERCISE  XVI 
Prove  the  following  relations  : 

l.    sin  70°  +  sin  50°  =  V3  cos  10°. 

2      sin80-sin60  =  tan^      3.     sin  2  0  +  sin  6  0 
cos  8  6  +  cos  6  6  cos  2  6  +  cos  6  6 

sinSg-rintf  8g       4g- 


sin  2  ^  +  sin  2  fl  =  fan    ^      ^  ^  _  B 

sml  A-  sin  2  5 


K.)L'  „.  PLANE   TRIGONOMETRY 


cos  0  —  cos  20  2  *   cosJ.+  cos 


cos  B  —  cos  ^4.  2 

9.    sin  (  J.  +  JB)  +  cos  (A  -  J5)  =  2  sin(45°  +  5)  cos  (45°  - 
cos  b  A  —  cos  3  ^4  ,   cos  2  A  —  cos  4  ^4  si  n  A 


10. 


sin  <)  A  —  sin  3  .A      sin  4^4  —  sin  ^  A          cos  4  ^4  cos  3  A 

11.  sin  (60°  +  A)-  sin  (60°  -  A)  =  .sin  A 

12.  cos  (30°  -  0)  +  cos(30°  +  0;  -  V^  cos  0. 


13. 


14  sin  6  +  sin  3  0  +  sin  5  0  +  sin  7  0  =  t      4  ^ 
cos  0  H-  cos  30  +  cos  50  +  cos  7  0 

15  sin  0  -  sin  50  +  sin  9  0  -  sin  18  0  =  cot  ±  Q 
cos  0  —  cos  5  0  —  cos  90  +  cos  13  0 


16_ 


sin  x—  sin  ^/ 


cos  x  -\-  cos  v          ^  ^ 

17.  —  £  =  cot  —  —  '   cot 

cos  x  —  cos  y  2  2 

18.  cos  3  0  +  cos  5  0  +  cos  7  0  +  cos  150=4  cos  4  0  cos  5  0  cos  6  0. 
19. 


20.   sin  50°  +  sin  10°  -sin  70°  =  0. 

2 


sin  (3  A  +  i?)  +  sin(J.  —  3  ^ 

22.  sin  80°  +  sin  70°  -  sin  10°  -  sin  20°  =  -t  sin  40°  4-  sin  50°. 

23.  cos  x  +  cos  2  #  +  cos  4  #  +  cos  5  #  =  4  cos  ~  cos  —  cos  3  r. 


RELATIONS   BETWEEN   TWO   OR  MORE   ANGLES       103 

24.  sin  O  +  P  +  7)  +  sin  (a  -  yS  -  7)  +  sin  (a  +  /3  -  7) 

-f-  sin  («  —  @  -f  7)  =  4  sin  a  cos  /?  cos  7. 

25.  sin  2  a  +  sin  2  /3  +  sin  2  7  —  sin  2  (a  -f  ft  -f-  7) 

=  4  sin  (/3  +  7)  sin  (7  +  a)  sin  (a 


=  cos  3  6 
cos  3  6  +  2!  cos  5  6  +  cos  7  0  ~  cos  5  0 


sin  30  +  2  sin  50  +  sin  10        .    c 
27.  __^=sm5 

S1H0  +  2  sin  30  +  sin  00 

sin     J  +  #  -  2  sin  A  +  sin  Qi  - 


28. 


cos  (^4  +  ^)  —  2  cos  J.  +  cos  (  A  —  B} 
29. 

cos(  — 


sin  (x  +  y  -|-  2)  +  sin(  —  ^  +  y  +  ^;  —  sin  (2:  —  y-\-z)+  s\ 

=  cot  ^. 
so.   cos  20°  +  cos  100°  +  cos  140°  =0. 

78.  The  product  of  two  sines,  of  two  cosines,  or  of  a  sine  and  a 
cosine  expressed  in  the  form  of  an  algebraic  sum. 

In  (1),  (2),  (3),  and  (4),  Art.  77,  the  u-v  formulas  are  ex- 
pressed in  a  form  which  is  quite  as  important  as  that  already 
considered,  and  which  is  so  convenient,  and  of  such  frequent 
application  that  the  formulas  are  here  reproduced  in  that  form. 
Using  the  left  for  the  right  and  the  right  for  the  left  members, 
they  are 


2  sin  oc  cos  y  =  sin  (&  +  y}  +  sin  (oc  -  y)  ;  (1  ) 

2  cos  ac  sin  y  =  sin  (x  +  y}  -  s  :n  (&  -  y}  ;  (2) 

2  cos  w  cos  y  =  cos  (oc  +  y}  +  cos  (x  —  y}  ;  (3) 

—  2  sin  x  sin  y  =  cos  (a?  +  y)  -  cos  (x  —  y).  (4) 

These  formulas  are  the  converse  of  the  u-v  formulas,  and 
may  be  conveniently  referred  to  by  that  name.  The  two 
groups  taken  together  are  useful  in  solving  problems  and  in 
performing  investigations  which,  without  them,  could  be 
handled  only  with  the  greatest  difficulty. 


104  PLANE   TRIGONOMETRY 

EXERCISE   XVII 

1.  Express  in  the  form  of  a  sum  or  difference  2  sin  6  0  sin  4  6. 

2  sin 60  sin 4  0  =  -  (cos(60  +  40)&cos(60  -  40)) 
=  -  (cos  100  -cos  20) 

=  00820- COS  100, 

2.  Express  in  the  form  of  a  sum  or  difference  cos  (A  —  2B) 
sin  (04 +  2  B). 

cos(^  -2B)sin(A  +25)=  |  (sin  2/1  -  sin  (-45)) 
=  £  (sin  2^4  +  sin  45). 

3.  Find  the  value  of  2  sin  75°  sin  15  °. 

2  sin  75°  sin  15°  =  cos  (75°  -  15°)  -  cos  (75°  +  15°) 
=  cos  60°  -  cos  90° 

=  i-o 

=  *• 

Express  as  a  sum  or  difference  the  following: 

4.  2  sin  60  cos  26.  0        30 

8.     COS  -  COS . 

5.  2  cos  40  sin  '20. 

6.  cos ? sin  ?*.  9-    2 sin  (2 ,1  +  10  cos  (A-*). 

2         2 

-*       **  10.    2  cos  3  J.  cos  01  — 2*). 

o  u        7  " 

1TC   "T*  11.    sin  (60°  +  0)  cos  (60° -  0). 

Prove  the  following  identities : 

12.  cos  (120°  +  0)  cos  (120°  -  0)  =  £  (2  cos  2  0  -  1). 

13.  cos  (30°  -  0)  cos  (60°  -  0)  =  |  (2  sin  20  +  V 3). 

14.  sin  (120°  -  0)  cos  (60°  +  0)  =  J-  (sin  60°  -  2  0). 

15.  sin  (0  +  45°)  sin  (0  -  45°)  =  -  |  cos  2  0. 

16.  cos  3  0  sin  2  0  -  cos  4  0  sin  0  =  cos  2  0  sin  0. 

17.  sin  3  0  sin  6  0  +  sin  0  sin  2  0  =  sin  4  0  sin  5  0. 

18.  sin  20  cos  0  +  sin  6  0  cos  0  =  sin  3  0  cos  2  0  +  sin  50  cos  20. 

19.  cos  (40°  -  0)  cos  (40°  +  0)  +  cos  (50°  +  0)  cos  (50°  -  0)  = 
cos  2  0. 


RELATIONS   BETWEEN    TWO   OR   MORE   ANGLES       105 

20.  sin  A  cos  {A  +  B)  —  cos  A  sin  (A  —  B)  =  cos  2  -4  sin  B. 

3  7T  4  7T    .  47T,  10  7T         A 

21.  2  cos  —  -  cos  —  -f  cos  —  -  -f-  cos  —  —  =  0. 

22.  4  sin  A  sin  ^  sin  Q  =  sin  (5  +  (7—  J.)  -f  sin  ((7+  .A  —  B) 

+  sin  (  j.  +  B  -  Q)  -  sin  (A  +  ^  -f  6Y). 


cos  3  ^4.  sin  2  A  —  cos  4  J.  sin  A    _  _      t  o  >4 
cos  5  A  cos  "2  A  —  cos  4  ^4.  cos  3  .A 

24.    4  sin  0  sin  (60°  +  0)  sin  (60°  -  (9)  =  sin  3  0. 


25.  4  cos  0  cos          +  0  cos     Z  -  0  =  cos  3  6. 

v  8         /       \  o 

26.  sin  20°  sin  40°  sin  80°  =  \  V3. 

27.  cos  20°  cos  40°  cos  80°  =i. 


CHAPTER   IX 
FUNCTIONS  OF  MULTIPLE  AND  SUBMULTIPLE  ANGLES 

79.  Functions  of  an  angle  in  terms  of  functions  of  half  the  angle. 
If  in  the  addition  formulas,  Arts.  69,  71,  73,  and  74,  we  put 
x  =  y,  we  have 

sin  (x  4-  x)  =  sin  x  cos  x  -\-  cos  x  sin  a?, 

cos  (x  +  x)  =  cos  x  cos  x  —  sin  x  sin  #, 

tan  (*  +  *)=   «*"  *  +  **"*, 
1  —  tan  x  tan  # 

and  c 


cot  x  -f-  cot  x 
sin  2  x  =  2  sin  a?  cos  a?  5  (1) 

cos  2  «  =  cos2  a?  -  sin2  a?;  (2) 


;  (3) 

(4) 

In  these  formulas  2#  may  have  any  value  whatever;  or,  in 
other  words,  2  #  is  any  angle  whatever. 

Hence,  these  formulas  are  to  be  regarded  as  formulas  for 
expressing  the  values  of  functions  of  an  angle  in  terms  of 
functions  of  half  the  angle.  They  may  also,  of  course,  be  re- 
garded as  formulas  for  expressing  the  functions  of  twice  an 
angle  in  terms  of  functions  of  the  angle  itself. 

80.  If  we  let  2x=  0,  we  have  the  formulas  in  the  following 
useful  form  :  ~  ^ 

sin6  =  2  sin  -  cos  -;  (1) 

10<5 


MULTIPLE   AND   SUBMULT  1PLE   ANGLES  107 

cos  0  =  cos2  -  -  sin2  -  (2) 

2  2 


2cos2--l. 


2tan- 

(3) 


1  -  tan2  - 
2 

cot2  -  -  1 
cot  6  = £-• 

2cot- 
2 

81.   Functions  of  an  angle  3  &  in  terms  of  functions  of  oc. 

If  in  the  addition  formulas  we  put  y  =  2  x,  we  obtain  expres- 
sions for  the  value  of  functions  of  3  #  in  terms  of  functions  of 
#,  as  follows : 

sin  (a;  +  2  a;)  =  sin  x  cos  2  x  4-  cos  x  sin  2  x 

=  sin  x  (cos2  a;  —  sin2  a:)  +  cos  a:  •  2  sin  a:  cos  a; 
=  sin  x  (1  —  2  sin2  a;)  +  2  sin  x  (1  —  sin2  a;) 
=  sin  x  —  2  sin3  a,1  +  2  sin  a:  —  2  sin3  a:. 

.-.  sin  3  3C  =  3  sin  a?  —  4  sin3  cc.  (1) 

In  like  manner, 

cos  (a;  +  2  a:)  —  cos  a:  cos  2  a;  —  sin  x  sin  2  a: 

=  cos  a:  (cos2  a;  —  sin2  a:)  —  sin  x  •  2  sin  a?  cos  a: 
=  cos  a:  (2  cos2  a;  —  1)  —  2  (1  —  cos2  a;)  cos  a; 
=  2  cos3  x  —  cos  x  —  2  cos  x  +  2  cos3  a:. 

.•.  cos  3  a;  =  4  cos3  x  —  3  cos  a%  (2) 


Also,  ten 3*=  - 


1  -  tan  z  tan  2  x      ^  _  fcftn  ^    2  tan  x 


1  —  tan2  a; 
3 


3  tan  a?  ^tan3  x  XON 

/.  tan  3  a?  =  -  -  .  (3) 


In  a  similar  manner  it  is  possible  to  obtain  formulas  for  the 
functions  of  higher  multiples  of  x  in  terms  of  functions  of  x. 


108  PLANE    TRIGONOMETRY 

82.   Functions  of  an  angle  expressed  in  terms  of  functions  of 
twice  the  angle. 

Since  cos  2  x  =  1  -  2  sin2 x, 

we  have  2  sin2  ;r  =  1  —  cos  2  x. 


.•.  sin  a?  = 


Also,  cos  2  x  =  2  cos2  a:  —  1, 

2  cos2  s  =  1  4-  cos  2*. 


1  +  cos2ag  *2\ 

Dividing  (1)  by  (2)  we  have 

(3) 


These  formulas  are  often  given  in  the  following  form,  where 
0 


*=-• 


(5) 


In  this  form  they  are  to  be  regarded  as  formulas  for  express- 
ing the  values  of  functions  of  a  half-angle  in  terms  of  functions 
of  the  angle  itself. 

The  magnitude  of  the  angle  determines  which  of  the  two 
signs  preceding  the  radical  is  to  be  employed. 

EXERCISE   XVIII 

1.  If  sin  6  =  1,  find  sin  2  9  and  sin  3  0. 

2.  If  sin  0  =  \,  find  cos  2  d  and  cos  3  0. 

3.  If  cos  6  =  |,  find  sin  20  and  cos  30. 

4.  If  tan  6  ==  l,  find  tan  2  0  and  tan  3  0. 

5.  If  tan  6  =  1,  find  sin  2  0  and  tan  3  6. 


MULTIPLE   AND   SUBMULTIPLE   ANGLES 

Prove  the  following  identities  : 
6.  cos4  0  —  sin4  0  =  cos  2  0.  cot  0  —  tan  0 


1 


a                                          an                 COt  0  +  tail  0 

7.     tan  0  4-  cot  0  =  2  esc  2  0.                                                           )t 

j 

^T 

tan  -2  0                               cot2  0-1 

H5j 

•T 

^ 

The  next  six  equations  are  especially  important,  and  may^e 
regarded  as  standard  formulas. 

1 

11     fSin64    DS6Y      11  -in  9     18        r*e      2  ~  sec2  *           s\\ 

^ 

'  ^"g  h    *ij  -                                               sec20     • 

12.  tan  9^     8in26                              ;„,*      sec0-.l 

•J 

l  +  cos29                            1  2-    2sec0  • 

l» 

TO          *  A             S*n  ^  "                                                       A                      f           A\ 

,    XI 

i           0oA*                   2O                        —  tanf77"-!-      1  2DI1 

i  T\r 

1  —  COS  2  U                             ^u-      ^             •       /i  —  ttl11  I     i     1    "c\  I'      i 

p 

1  -  sm  0             \4      2/  ^|V  <£* 

9      i  _  eos  Q 

} 

1  4-      t*in       —                          •                                                                                             i    i     Z>         ;      *** 

1 

1 
6 

2          sin  9                                   •  \         &        ta"V9 

?- 

-| 

i 

,  9  _  1  4-  cos  9                           eos  (Jby  ^/\         ^\^jf 

°  2         sin  9                                                                      2 

/A           e\2                                 cos  20     _,       ,,ro     ^ 

16.    [  sin  cos  —  ]  =1  —  sin  9.   ^        i   i  ^i     o  /} 

4 

\      "           */                                                                                  ^    s^^  \ 

\* 

•        /i                                 £&i 

/->       o0      1  -h  sec  0                       sin  3  0      cos  30      /-» 

* 

2         sec  0                           sin  0         cos  0 

11 
^ 

1  —  cos  J.  +  cos  B  —  cos  (A  +  ./?)            ^4  ^^-B 

& 

1  +  cos  4  —  cos  B  —  cos  (  A  +  .#)          "  2  ~      2 

r 
~j 

25.    tan  (45°  +  0)  +  tan  (45°  -  0)  =      2^0  ' 

N^ 

26.    tan  2  0  -  sec  0  sin  0  =  tan  0  sec  2  0. 

^x 

v                                                                                                                i 

^*. 

1 

\ 

T*, 

siii2«-sm2/3                Hn/rt  +  £N                        T 

0 

*»«         .                                    •        /i              /->  """    Lail  V        1     r*x 

sin  a  cos  a  —  sin  p  cos  p                                                     **  \ 

•f 

i 

OQ     cos  0  +  sin  0      cos  0  —  sin  0  _  9  ^  o  ^ 

•h 

'-O 

cos  0  —  sin  0      cos  0  +  sin  0                                              i' 

V  ^ 

-  L 


110  PLANE    TRIGONOMETRY 

2g     cos(0  +  15°)  _  sin  (9  -  15°)  =      4  cos  2  0 
sin  (0  +  15°)      cos  (6  -  15°)      1  +  2  sin  20 


30. 


32 


sin  20  +  sin  0 

;-pos:;+si"^=ta, 

1  +  cos  20  +  sin  20 

-  tan  0  +  1  =  1  -  sin  2  0 
tan  0  +  1  cos  2  0 


33  sin  2  0        1  -  cos  0  =         0 
1  -  cos  2  0  "    cos  0  '"  2  ' 

34  sin  (n  +  1)0  +  sin  (ft-  1)0  +  2  sin  w0  =  cot  0 

cos  (n  —  1)0  —  cos  (w  +1)0  2 


35. 


. 

COS  0  —  Sill  0 

36.    sin  6  0  +  sin  4  0  -  sin  2  0  =  4  sin  2  0  cos  0  cos  3  0. 


37.  (sec  20  +  1)  Vsec2  0  -  1  =  tan  2  0. 

38.  4  cos  0  cos  (60°  -  0)  cos  (60°  +  Q  =  cos  3  #• 

39.  16  cos  20°  cos  40°  cos  60°  cos  80°  =  1. 


40.    tan  (45°  +  0)  =  X/f 

M  —  sin  2  0 

41  sin  (n  +  1)0  -  sin  (n  -  1)0  _  t      0. 

cos  (ft  +  1)  0  +  cos  (n  —  1)  0  +  2  cos  w0  ~~         2 

42.    cos2  O  +  1)0  -  cos2  nO  =  -  sin  (2  n  +  1)0  sin  0. 

83.   Identities  that  are  true  for  angles  whose  sum  is  180°  or 

90°.  When  three  angles  are  involved  whose  sum  is  either  90° 
or  180°,  many  relations  are  found  to  exist  that  do  not  hold  true 
for  angles  in  general. 

For,  if  A  +  B  +  C=  180°,  we  have  (Art.  53,  p.  78), 

sin  (A  +  B)  =  sin  C,  cos  (^4  +  B)  =  -  cos  (7,  tan  (A+B)=-  tan  (7, 


MULTIPLE   AND   SUBMULT1FLE   ANGLES  111 

and  similar  relations  hold  between  functions  of  the  sum  of  any 
two  of  the  given  angles,  and  the  corresponding  functions  of  the 
third  angle,  since  the  sum  of  any  two  is  the  supplement  of  the 
third. 

Also,  if  —  (-  —  +-  =  90°,  the  sum  of  any  two  of  these  angles 

is  the  complement  of  the  third.     Therefore, 
in  (I  +  f  )  =  cosf  ,  cosg  H-  f  )  =  sinf  ,  tan  (f  +  f)=  cotf  , 


sn 


and  similar  relations  hold  between  functions  of  the  sum  of  any 
two  of  the  angles  and  the  corresponding  co-functions  of  the 
third. 

Ex.  1.  If  A  +  B  +  O=  180°,  prove  that 

sin  2  A  -f  sin  2  B  —  sin  2  0  =  4  cos  A  cos  B  sin  O. 

Left  member  =  2  sin  (A  +  5)  cos  (4  -  B)  -  2  sin  C  cos  C 
=  2  sin  C  cos(.4  -  £)  +  2  sin  C  cos  (A  +  £) 
=  2  sin  C  [cos  (.1  +  B)  +  cos  (.4  -  5)] 
=  2  sin  C  (2  cos  A  cos  7?) 
=  4  cos  -4  cos  R  sin  C. 

Ex.  2.    If  A  -h  5  +  #  =  180°,  prove  that 

ABO 

cos  A  +  cos  #  +  cos  (7=  1  +  4  sin  —sin  —sin  —  . 


Left  member  =  2  cos   --    cos  -        +1-2  sina 


/^»  A  T>  SI 

=  1  +  2  sin     cos  ^-         -  2  sm2. 


=  1  +  2  sin 


112  PLANE   TRIGONOMETRY 

Ex.3.    If  A  +  B+  (7=180°,  prove  thai 

tan  A  +  tan  B  +  tan  (7  =  tan  A  tan  B  tan  0. 
Since  A  +  B  =  180°  -  C,  tan  (^4  +  5)  =  -  tan  C ; 

tan  A  +  tang    =  _  tftn  c< 
1  —  tan  /I  tan  B 

Clearing  of  fractions,     tan  A  +  tan  B  =  —  tan  C  +  tan  A  tan 
.-.  tan  .4  +  tan  B  +  tan  C  =  tan  /I  tan  B  tan  C. 


EXERCISE  XIX 

If  A  +  B  +  C=  180°,  prove  that 

1.  sin  2  A  4  sin  2  B  +  sin  2  C  =  4  sin  A  sin  B  sin  C. 

2.  cos  2  A  -f-  cos  2  .B  4  cos  2  (7=  —  1  —  4  cos  A  cos  .B  cos  0 

3.  cos  2  A  -  cos  2  £  4  cos  2  (7=  1  -4  sin  A  cos  B  sin  (7. 

4.  sin  2  A  —  sin  2  B  —  sin  2  "(7=  —  4  sin  A  cos  B  cos  (7. 

A       .B       (7 

5.  cos,  A  +  cos  B  —  cos  0=  —1  +  4  cos  —  cos  —  sin  — . 

A       B       C 

6.  sin  A  4  sin  B  4  sin  (7=4  cos  —  cos  —  cos  — . 

7.  sin  A  -f  sin  I?  —  sin  C  —  4  sin  —  •mt  —  cos  — . 

i-         ^         '— 

8.  sin2  A  +  sin2  B  —  sin2  (7=2  sin  A  sin  .Z?  cos  (7. 

9.  cos2  A  +  cos2  5  -  cos2  (7=  1-2  sin  A  sin  .B  cos  (7. 

14  -    CW        -    '-  :  0jf  a,/;.: 

sin  A  4  sin  .B  —  sin  (7  A,      .Z?  *  A*<I  Pr*** 

10.  -  =  tan  —  tan— . 

sin  A  4  sin  B  4  sin  C  2 

sin  2  A  +  sin  2  J9  4  sin  2  (7 
sin  2  A  4  sin  2B  —  sin  2  f 

1  4  cos  A  —  cos  J5  4  cos  C  B     ,  C 

12.  —  =  tan  —  cot— . 

I  4  cos  A  4  cos  ^  —  cos  (7  22 


sin  J.  +  sin  5 -f- sin  (7 


*  - 


14.  cot  A  cot  B  4-  cot  jB  cot  (7  4-  cot  (7  cot  A 


• 


MULTIPLE   AND   SUBMULT1PLE   ANGLED  113 

M^iif, 

—  1. 


j»  7?        /7  (7       ^4. 

15.    tan  —  tan  —  +  tan  —  tan  -  +  tan  -  tan  —  =  1. 


A  7?  C*  ABC 

16.    cot  —  +  cot  —  4-  cot  —  =  cot  —  cot  —  cot  —  . 


17.    sin2 


ABO  A+  B       B+  0       0^ 

18.    cos  —  4-  cos—  4-  cos—  =  4  cos  —  —  cos  —  ^  —  cos  —  - 

2  2  "2  4  44 


19.    sin  (A  +  B-  (7) 


4-  C-  A)  +  sin  ((7+  A  - 
=  4  sin  A  sin  B  sin  0. 


.    sin  (^1  +  2  £)  +  sin  (B  +  2  (7)  4-  sin  ((74-  2' 


—  4  sm 


sin 


-C  •    C-A 

sin  —  -  — 


CONANT'S  TRIO.  —  8 


CHAPTER   X 
INVERSE   TRIGONOMETRIC   FUNCTIONS 

84.  If  sin  6  =  a,  where  a  is  any  known  quantity,  9  may 
have  any  one  of  an  infinite  number  of  values.  The  symbol 
"  sin"1  a  "  is  used  to  denote  the  angle  whose  sine  is  #,  and  is, 
accordingly,  read  "the  angle  whose  sine  is  a."  It  is  some- 
times called  the  inverse  sine,  or  the  anti  sine  of  a. 

To  illustrate  the  use  of  this  notation,  let  us  take  the  equation 

cos0  =  J.  (1) 

We  know  from  this  that  0  may  equal  60°,  300°,  420°,  660°,  .... 
To  state  the  fact  that  0  may  equal  any  one  of  these  angles,  we 

employ  the  equation  /,  _-,  ^ 

6  =  cos  1 1,  (2) 

which  is  read  "0  =  the  angle  whose  cosine  is  |." 
We  are  then  to  understand  that  (1)  and  (2)  are  inverse  state- 
ments, the  former  asserting  that  the  cosine  of  some  angle,  #,  is 
equal  to  ^,  and  the  latter  asserting  that  9  is  the  angle  whose 
cosine  is  |. 

From  (2)  we  also  understand  that  60°,  300°,  420°,  •••,  are 
angles  that  satisfy  the  equation,  since  the  cosine  of  each  of 
these  angles  is  J.  In  other  words,  (2)  is  satisfied  by  any  of 
the  angles  (Art.  64,  p.  87)  included  in  the  general  expression 

7T 

2^±f. 

Similarly,  if  tan  0=1, 

then  the  equation  9  =  tan'1  1 

asserts  that  9  may  equal  45°,  225°,  405°,  •••.     That  is,  9  may 
have  any  one  of  the  values  represented  by  the  expression 

+  ,    7T 

H7T  +  —  • 

114 


INVERSE    TRIGONOMETRIC    FUNCTIONS  115 

It  is  strongly  urged  that  the  student  become  familiar  at  the  out- 
set with  the  idea  that  the  expressions  sin"1  1,  cos"1  J,  tan~JV3, 
etc.,  are  single  symbols,  and  denote  angles.  They  represent 
angles  just  as  definitely  as  do  the  symbols  0,  <£,  A,  B,  x,  y,  etc., 
which  are  used  so  frequently  for  that  purpose.  The  only  point 
to  be  noted  is,  that  the  angle  which  is  represented  in  this  man- 
ner is  described  by  means  of  one  of  its  trigonometric  functions. 

85.  Angles    expressed    by  the    symbols    sin"1^,   cos'^VS, 
tan"1  1,  etc.,  are  called  inverse  trigonometric  functions,  or  in- 
verse circular  functions. 

Since  a  central  angle  has  the  same  magnitude  in  degrees  as 
the  intercepted  arc,  these  functions  are  used  to  represent  arcs 
as  well  as  angles.  The  notation  arc  sin  #,  arc  cos  J,  arc  tan 
JV3,  etc.,  is  often  used  instead  of  sin'1  a,  cos"1!,  tan~1^V8,  etc. 

In  using  the  notation  here  adopted,  the  student  should  note 
that  the  symbol  —  1  is  not  an  algebraic  exponent.  That  is, 

sin"1  a  is  not  the  same  as  (sin  a)"1. 
The  former  expression  denotes  the  angle  whose  sine  is  a,  and 

the  latter  denotes  -  ,  or  esc  a. 
sin  a 

86.  The  smallest  numerical  value  of  an  angle  whose  sine, 
cosine,  tangent,  etc.,  have  given  values,  is  called  the  principal 
value  of  the  angle. 

Thus,  the  principal  values  of 


are       30°,  ±120°,  -45°,  60°. 

In  a  case  like  the  second,  where  two  values  are  given,  which 
are  numerically  equal  but  have  opposite  signs,  the  positive 
value  is  usually  understood.  Thus,  the  principal  value  of 
cos-i(_l)  is  usually  considered  to  be  120°. 

To  avoid  ambiguity,  it  will  be  understood  that,  when  any  of 
these  symbols  are  employed,  the  principal  values  of  the  angles 
are  referred  to. 

If  a  is  positive,  the  principal  values  of  all  the  inverse  func- 
tions except  vers"1^  and  covers"1  a  lie  between  0°  and  90°. 
The  principal  value  of  vers"1^  lies  between  0°  and  180°,  and  the 


116  PLANE   TRIGONOMETRY 

principal  value  of  covers"1  a  lies  between  0°  and  90°,  or  between 
180°  and  270°. 

If  a  is  negative,  the  principal  values  of  sin"1  a  and  csc"1^  lie 
between  0°  and  —  90°,  or  between  180°  and  270°.  The  principal 
values  of  cos"1  a  and  sec"1  a  lie  between  90°  and  180°,  or  be- 
tween —  90°  and  —180°.  The  principal  values  of  tan"1  a  and 
cot"1  a  lie  between  90°  and  180°.  As  stated  above,  the  positive 
values  of  these  angles  are  usually  employed.  Since  vers  0  and 
covers  6  are  always  positive,  vers~1a  and  covers'1  a  are  impossi- 
ble when  a  is  negative. 

87.    Ex.  1.     Prove  that  sin"1!  +  cos"1  if  =  cos'1!  f.  (1) 

Let  sin-1  f  =  a,  cos-1  1|  =  ft,  cos-1  f  f  =  y. 

Then,  sina=f,  cos/3={f,  cosy=ff. 

We  are  to  prove  that  a  +  ft  =  y.  (2) 

This  can  be  done  by  proving  that  any  function  of  a  +  ft  is  equal  to  the 

same  function  of  y,  since,  if  two  sines,  two  cosines,  two  tangents,  ••-,  are 

equal,  the  principal  values  of  the  angles  are  also  equal. 

In  this  case  we  select  the  cosines  ;  and  we  are  now  to  prove  that 

cos  («  +  /?)  =  cosy.  (3) 

Expanding,  cos  «  cos  ft  —  sin  a  sin  ft  =  cos  y.  (4) 

The  values  of  cos  ft,  sin  ct,  and  cos  y  are  already  known  ; 
3    and,  obtaining  the  values  of  cos  a  and  sin  ft  from  the  figures 
in  the  margin,  and  substituting  in  (4),  we  have 


t*  -  *§     =  li 

.•.  cos(«  +  ft)  =  cosy. 


Ex.  2.     Prove  that  cos'1!  +  sin"1  1\  +  sin'1  I 

Let  cos-1  1  =  a,  sm-1fs  =  p,  si 

Then,  cos  a  =  f,  sin/8  =  T\, 

We  are  to  prove  that    a  -f  ft  +  y  =     , 


Selecting  in  this  case  the  sines,  we  proceed  as  follows: 


=  cos  y. 
sin  a  cos  ft  -f  cos  a  sin  ft  =  cos  y. 


INVERSE   TRIGONOMETRIC   FUNCTIONS 

Substituting  numerical  values,  we  have 


117 


Ex.  3.    Prove  that 


2  sin"1  —  —  +  tan"1  --  cos"1  —  -  =  0. 
VlO  ?  V2 


Let 

Then, 

We  are  to  prove  that    2  a  +  fi  —  y  =  0, 


v/10  7 

sin  a  =  -  ,     tan  B  =  -  , 
VlO  7 


V2 

cos  y  = 

V2 


or,  2  a  +  j8  =  y. 

Selecting  the  tangents  as  convenient  functions  to  deal  with  in  this  case, 
we  proceed  as  follows  : 

tan  (2  a  +  ft)  =  tan  y 


1  - 


The  left  member  =  tan  2  «  +  ten  fl  < 
1  -  tan  2  «  tan  3 


1  - 


1-    2ta"«    tan/? 
l-tan2« 


But 


tan  y  =  1. 
.•.  tan  (2  a  +  /#)  =  tan  y. 

2  «  +  ^8  -  y, 


63 


Ex.  4.    Prove  that 

2  sin-1  -A-  +  cos"1  ~  +  i  tan-1^  =  TT. 

Via          ^52        7 


, 

65  7 

2  16  24 

Then,       sin«=—  '=,  cos  (3=—,  tan  y  =  ±- 

via  6o  7 


V13 
sin« 

We  are  to  prove  that 


2  ce  +  ^8  -  TT  -  |  y. 


118  PLANE   TRIGONOMETRY 

Selecting  the  sines  as  the  most  convenient  functions  with  which  to  work 
in  this  case,  we  proceed  as  follows: 

sin  (2  a  +  /3)=  sin  (TT  -  \  y). 
sin  2  a  cos  ft  +  cos  2  a  sin  ft  =  sin  \  y.  (1) 

All  the  functions  of  a,  ft,  and  y  can  be  determined  at  once  from  the 
proper  figures  ;   and  the  values  of   sin  2  a,  cos  2  ft,  and  sin  |  y  must  be 

computed.  2          3         12 

sin  2  a  =  2  sin  a  cos  a  =  2  •  —  __  •  —  __  =  -- 
V13     V13       13 

945 

cos  2  a  =  cos2  «  —  sin2  a  =  ---  =  — 
13      13      13 


Substituting  in  (1),  we  have 

tt-tf  +  A-ti=!» 

192  +  315  =  3 
845  5' 

*  =  *• 

2  a  +  /3  =  TT  -  \  y. 
$y  =  TT. 

EXERCISE   XX 
Prove  that 

1.  sin"1  ^  =  008-!  if.  3.    cos"1  3f  =  esc"1  ff. 

2.  sin-i  T\  =  tan-i  T52  .  4.    sin-i-=2sin-1--L 

5  VlO 

5.    tan-i  J-tan-i}  =  tan-il. 

^N      6.    sin'1  if  +  cos"1  if  =  sin-1  f  . 

7.  sin-if-f  tariff  =tan-iff 

8.  tan-1  T2T  +  cot-1  -2Y4-  =  tan'1  J. 

9.  tan-i^  +  tan-i|=  sin-ii. 

«  o  V2 

—      10.    sin'1  —  =  +  tan'1  -  =  cos"1  -—  - 
V5  3  V2 

11.  COt-1  f  I  +  Cot"1  J/-  =  COt-1  1. 

12.  2  tairi  1=00^1^2. 

13.  2  tan-1  £  +  tiiir1  J  =  tan-1      . 


\\ 


INVERSE   TRIGONOMETRIC    FUNCTIONS  119 

14.    tan'1 1  +  cot'1  4  =  |  cos-1 f . 
is.    sin-if  +  cot-i|-tarriJU|. 

16.    tan"1  |  +  tan-1  ^  =  tan"1 1  +  tan"1  ^. 

18.  2  cos-1  If  =  tan-iifl. 

19.  cos"1  x  —  2  cos"1  \/  - 

\     f) 

20.  tan"1  x  -f  tan"1  y  —  tan": 

_ 1  —  xy 

\J\ 21.    tan'1  x  +  cot-1  O  4-  1)  =  tan-1  (y?  +  x+  1). 

22.    tan'1  -  -  tan-1  ^-^  =  -  • 
y  x+  y       4 


23.    sin  l  a  +  cos"1  b  =  cos  -1  (b  Vl  —  a2  — 


24.    tair1  - — \  +  tan-1  ^ — ^  -f  tan'1  - — -  =  0. 
1  +  ab  1  -f  bo  1  -}-  ca 


25.    sin  (2  sin-1  a)  =  2  a  Vl  -  a2. 


26.  sin  f  cos"1  -  J  =  tan  f  sin"1 

\  O/  V 

27.  sin  (sin-1  a  +  sin"1  b)  =  a  Vl  —  b2  4-  £  Vl  —  a2. 

28.  tan  (tan"1  a  -f  tan"1  5)  = 


1- 


29.    tan  (2  tan"1  a)  =  — - 


30.    cos  (2  tan-1  j)  =  sin  (4  tan'1 1). 

88.   Solution  of  equations  expressed  in  the  inverse  notation. 

The  method  of  solution  of  equations  that  are  expressed  in  terms 
of  inverse  functions  is  best  illustrated  by  means  of  examples. 
Ex.  l.    Solve  the  equation 

tan-1  (x  +  1)  +  tan-1  (x  -  1)  =  tan'1  ^-. 

Let  tan-1  (*  +  !)  =  «,  tan-1  (*-!)  =  #  tan"1  ^  =  y. 

Then,  tan«  =  a;+l,  t&u  fi  =  x  —  1,  tany  =  ^v. 

To  find  what  values  of  x  will  satisfy  the  equation 


120  PLANE   TRIGONOMETRY 

when  tan  a,  tan  (3,  and  tan  y  have  the  above  values,  we  proceed  as  follows  : 
tan  (a  +  /?)  =  tan  y. 

Then  the  left  member  =    *an  a  +  tan  ft 

1  —  tan  a  tan  p 


2-z2 
Equating  this  to  tan  y,  we  have 

2x     =  8 
2-z2     31' 


Solving,  we  have  a:  =  J,  or  —  8. 

The  second  value  is  inadmissible  as  long  as  we  use  the  principal  value  of 
the  angles.     Therefore,  x  _  l 

Ex.  2.    Solve  the  equation 

tan"1  x  +  tan"1  (1  —  x}  =  2  tan"1  V#  —  x2. 


Let  tan-^^a,  tan"1  (1  -  x)=  (3,  tan-Vz  -  x*  =  y. 

Then,        tan  a  =  x,  tan  /?  =  1  -  ar,  tan  y  =  Vz  -  x2. 

To  find  what  values  of  x  will  satisfy  the  equation  we  proceed  as  follows  : 
tan  (a  +  ft)  =  tan  2  y, 

tan  ft  +  tan  /3    _    2  tan  y 
1  —  tan  a  tan  /2      1  —  tan2  y' 


l_a;(l-a;)      1  -  x  +  x2' 

1  =  2V^^2. 

Solving,  x  =  %. 

EXERCISE   XXI 

Solve  the  following  equations  : 

2.  sin"1^  =  cos"1(—  x). 

3.  tan"1  x  =  cot"1  x.  _,  1      2?r 

7.    tan  *  x  4-  ^  tan  x  -  =  — — 

4.  tan"1  a:  =  cot"1  (  —  a?) .  ^ 

-  +  sm     —  =  2'  :~T' 


Ml 


INVERSE   TRIGONOMETRIC   B^UNCTIONS 

9.    tan"1  (x  4-  1)—  cot"1 -  =  tan~1-. 

x  —  1  *j 

10.  tan"1 2  a;  4- tan-1 3  a  =  —  • 

11.  cos"1  x—  cos"1  VI  —  x2=  cos"1  a; VS. 

12.  sin"1  (3  x  —  2)  4-  cos"1  x  =  cos"1  VI  —  x2.  \  *  -"S 


121 


13. 


x-2 


2      4 


14. 


"2  5  x 

15.  sin  (cot"1  J)=  tan  (cos"1  Vie). 

16.  tan  (cos"1  x)  =  sin  (cot"1  J  )  . 


17.    sin"1  -  =  sin"1  -  4-  sin"1  -  • 
x  a  o 


18.       tan"1   -7- =  tan-1  (cos 
Vsm  a? 


19.    esc"1  x  =  sec"1- 


COS 


20. 


21.    tan"1  ^±4  4-  tan  - — i  =  tan~1(-7). 


;-  /«-/    <V, 

/( 1 


>>;*>;•»  4 


>l      1    I  T- 


'  1 1  < ! 


CHAPTER   XI 

THE  GENERAL   SOLUTION  OF  TRIGONOMETRIC   EQUA- 
TIONS 

89.  A  trigonometric  equation  is  an  equation  in  which  the  un- 
known quantity  or  quantities  appear  in  the  form  of  trigono- 
metric functions. 

These  equations  have  been  used  with  the  utmost  freedom  in 
previous  chapters,  though  no  formal  definition  has  been  given 
until  the  present  time.  They  have  been  used  in  many  differ- 
ent ways,  involving  one  or  more  functions,  one  or  more  angles, 
and  one  or  more  values  of  the  given  angles  in  any  single 
equation. 

At  first  the  only  angles  used  were  acute  angles,  and  an  equa- 
tion was  understood  to  involve  functions  of  an  acute  angle 
only.  Then  the  idea  was  introduced  of  an  angle  unrestricted 
in  magnitude ;  and  after  this  had  been  done,  all  results  were 
freed  from  the  restraints  which  had  previously  been  imposed 
by  the  fact  that  we  were  dealing  with  acute  angles  only. 

A  large  class  of  the  equations  with  which  we  have  previously 
been  concerned  consist  of  trigonometric  identities,  that  is,  equa- 
tions in  which  both  sides  had  the  same  value  for  all  possible 
values  of  the  angles  employed,  though  the  form  might  be 
different. 

Examples  of  these  are  the  formulas  that  have  been  proved 
from  time  to  time,  as,  sin2  6  +  cos2  0  —  1 ;  sin  (x  -f  «/)  =  sin 
x  cosy  +  cos  x  sin  y ;  etc.  Equations  of  this  kind  are  true  for 
all  possible  values  of  the  angle  or  angles  involved. 

But  trigonometric  equations  are,  of  course,  not  ordinarily 
true  for  all  values  of  the  angles  involved.  For  example,  if  we 
consider  the  equation  cos  Q  _.  \ 

we  see  at  once  that  we  can  assign  but  two  values  of  6  between 
0°  and    360°   that    satisfy    this    equation.       In    other   words, 

122 


GENERAL  SOLUTION  OF  TK1GONOMETIUC  EQUATIONS     123 

cos  6  =  £  is  true  only  for  0  =  60°  and  6  =  300°,  as  long  as  0  is 
restricted  to  values  between  0°  and  360°.  If  angles  of  unre- 
stricted magnitude  are  allowed,  cos  6  =  J  is  satisfied  by  all 
values  of  6  that  are  included  in  the  general  expression 

0-Siirdbg, 

O 

and  by  no  other  values. 

In  like  manner,  the  equation 


is  satisfied  by  all  values  of  0  that  are   given  by  the  general 
expression 

0=W7T+^, 

o 
and  by  no  other  values  ; 

sin  6  =  J  V2 

by  those  values  of  6  that  are  given  by  the  expression 


and  by  no  other  values  ;  and  so  on  for  other  examples  that 
might  be  given.  In  all  these  illustrations  it  is  to  be  under- 
stood that  n  is  any  positive  or  negative  integer  or  zero. 

The  solution  of  an  equation  is  the  determination  of  the  value 
of  the  angle  or  angles  that  satisfy  the  equation.  In  Art.  67, 
p.  88,  a  method  of  solution  was  given  by  means  of  which  some 
of  the  simpler  forms  of  trigonometric  equations  could  be  treated. 
But  at  that  time  only  a  limited  number  of  the  formulas  of 
transformation  were  at  our  disposal.  Hence,  the  number  of 
classes  of  equations  that  could  be  handled  was  necessarily  quite 
limited. 

The  methods  of  reduction  and  transformation  that  are  now 
available  make  it  possible  to  solve  many  classes  •  of  equations 
that  were  formerly  quite  out  of  our  reach,  and  also  to  sim- 
plify some  of  the  methods  previously  employed.  The  present 
chapter  will  illustrate  some  of  the  simpler  cases  of  this  kind. 

This  work  should  be  looked  upon  as  an  extension  of  that 
given  in  Art.  68,  p.  90. 


124  PLANE   TRIGONOMETRY 

90.    Solution  of  equations  of  the  form 

a  cos  6  +  b  sin  6  =  c.  (1) 

A  simple  method  of  solving  equations  of  this  form  is  fur- 
nished by  the  introduction  of  what  are  termed  auxiliary  angles, 
as  follows  : 

Assume  a  right  triangle  whose  legs  are  a,  6,  and  designate 
by  (/>  the  angle  lying  opposite  the  leg  b.  The  hypotenuse  of 
this  right  triangle  is  Va2-}-£2,  and  we  now  have 

cos  <£  =  —  ,  and    sin  <f>  =  • 


Dividing  each  member  of  the  original  equation  by  Va2  -}-  52, 
we  have 

cos  0  + sin  6  =        °  (2) 


Substituting  cos  </>  and  sin  <p  for  their  respective  values  in  this 
equation  we  have 

cos  <j>  cos  0  H-  sin  <f>  sin  0  =  — :  G 


or,  cos  (0  —  <£)  = 


Since  a,  b,  and  c  are  known,  cos  (6  —  <£)  is  known,  and  9  —  <£ 
can  at  once  be  found  from  the  tables.  Calling  this  angle  «, 
for  convenience  we  have 

cos  (#  —  <£)  =  cos  a. 
.  •.     6  —  <t>  =  2  MTT  ±  a,   Art.  64,  p.  87. 
#  =  2  ftTT  +  ^  ±  a. 

The  cosine  -of  an  angle  can  never  be  numerically  greater  than 
unity.  Hence,  in  dealing  with  the  equation  cos  (0  —  </>)  -— 


it  is  to  be  remembered  that  we  must  have  c  ^  Va2+£2.  If 
c  >  Va2  -f-  b'2,  there  is  no  real  value  of  0  —  <$>  which  will  satisfy 
the  equation. 


GENERAL  SOLUTION  OF  TRIGONOMETRIC  EQUATIONS     125 

Ex.  i.    Solve  the  equation   V3  cos  0  +  sin  6  =  V2. 
Dividing  both  sides  of  the  equation  by  V3  +  1,  i.e.  by  2,  we  have 


In  this  case  we  have  a  =  V3,  6  =  1,  and  Va'2+  62  =  2.    Hence,  the  auxiliary 
angle  <f>  is  equal  to  30°.     The  original  equation  then  becomes 

cos  30°  cos  6  +  sin  30°  sin  0  =  |V2. 
cos(0-30°)=|V2. 

But  £\/2  is  the  cosine  of  45°.     Hence,  we  write 
cos  (0-30°)  =  cos  45°. 


4' 

6±4* 

Ex.  2.    Solve  the  equation     5  cos  6  4-  2  sin  0  =  4. 
In  this  problem  we  have  a  =  5,  and  6  =  2.     Dividing  both  sides  of  the 
equation  by  Va2  +  ft2,  we  have 

JL  cos  0  +  -?=  sin  0  =  -t=  •  (1) 

A/29  V29  A/29 

In  the  preceding  example  we  were  able  to  find  the  value  of  <£  from  the 
familiar  coefficients and  -,  which  we  already  knew  were  the  cosine  and 

sine  respectively  of  30°.     But  in  this  example  we  have  unfamiliar  values  to 
consider. 

From  the  figure  on  the  margin  of  the  page  we  see  that  <f>  is  an 

o  i^i 

angle  whose  cotangent  is  ~.  Turning  to  the  tables,  we  find  that  the 
o 

value  of  <f)  is  68°  12' ;  and  (1)  can  now  be  written 

cos  68°  12'  cos  0  +  sin  68°  12'  sin  0  =  -4=  • 

V29 

Letting  a  equal  the  angle  whose  cosine  is  — —  this  becomes 

V29 
cos  (0-68°  12')  =  cos  a. 

Reducing  the  value  of  — ^—  to  a  decimal,  we  find  it  to  be  0.7428 ;  and, 

V29 
consulting  the  tables,  we  find  that  the  angle  whose  cosine  is  0.7428  is  42°  2'. 

Therefore,  cos  (0  _  68°  12')  =  cos  42°  2', 

0  -  68°  12'  =  2  mr  ±  42°  2', 


NOTE.  Each  of  the  foregoing  examples  could  have  been  solved  by 
replacing  either  sin  0  or  cos  0  by  its  value  in  terms  of  the  other,  then 
obtaining  the  value  of  the  single  function  involved,  and  finally  obtaining 
the  value  of  0  from  the  value  of  this  function.  But  the  process  just  ex- 
plained is  much  simpler  and  better. 


126  PLANE   TRIGONOMETRY 

91.  Solution  of  equations  involving  multiple  angles.  The 
simplest  forms  of  equations  involving  multiple  angles  have 
already  been  considered  (Art.  68,  p.  90).  But  these,  and 
also  many  other  forms  of  equations  in  which  multiple  angles 
appear,  are  more  conveniently  treated  by  means  of  the  various 
reduction  formulas  that  are  now  available. 

The  following  problems  will  illustrate  some  of  the  methods 
of  most  frequent  application. 

Ex.  l.    Solve  the  equation  sin  3  6  -f  sin  7  6  =  sin  5  0. 
By  (5),  Art.  77,  p.  100,  we  have 

2  sin  50  cos  20  =  sin  50. 

.-.  sin  5  0  =  0,  or  cos  2  6  =  |. 
If  sin  50  =  0,  then  5  0  =  HIT. 


If  cos  2  0  =  -,  then  20  =  2  mr  ±  £ 


Therefore,  the  general  values  of  0  that  satisfy  the  equation 
sin  30  +  sin  7  0  =  sin  5  0, 

are  0  =  ^?,  and  0=  mr  ±  £. 

o  o 

Ex.  2.    Solve  the  equation     cos  4  0  —  cos  6  0  —  sin  20  =  0. 
Applying  the  proper  reduction  formulas,  we  have 
2  sin  5  0  sin  0  -  2  sin  0  cos  0  =  0. 
.-.  sin  0  (sin  5  0  —  cos  0)  =  0. 

From  the  first  factor  we  have 

sin  0  =  0. 
.-.  0  =  n7r.  (1) 

From  the  second  factor  we  have 

cos  0  =  sin  5  0 


=  cos   r  - 


.•.0=2mr±(|-50Y 
Using  the  positive  sign,  we  have 


12 


GENERAL  SOLUTION  OF  TRIGONOMETRIC  EQUATIONS     127 

Using  the  negative  sign,  we  have 


[the  sign  of  n  being  left  unchanged  because  n  denotes  all  negative  as  well 
as  all  positive  integers]  mr  t  IT 

*--j+g-  (3) 

Collecting  the  values  given  in  (1),  (2),  and  (3),  we  have  as  the  general 
values  of  0  that  satisfy  the  given  equation 


Ex.  3.    Solve  the  equation     cos  2  x  =  cos  x  +  sin  x. 

Expressing  cos  2  #  in  terms  of  functions  of  x  we  have 

cos2  x  —  sin2  x  =  cos  x  +  sin  x  ; 
(cos  x  +  sin  x)  (cos  x  —  sin  x)  =  cos  x  +  sin  x. 

.-.  cos  x  +  sin  x  —  0,  (1) 

or,  cos  a;  —  sin  ar  =  1.  (2) 

From  (1)  we  have  tan  x  —  —  1. 


From  (2)  we  have 


7T 

x  =  nit  —  — 


1  1.1 

—  cos  x sin  x  =  — -, 

V2  V2  V2 

or,  cos  ?  cos  x  —  sin  -  sin  x  =  cos  ?, 

444 


f        *\ 
cos  ar  +  —  )  =  cos 

V       4/ 


x  =  2  /ITT,  or  #  =  2  n?r  —  —  • 

EXERCISE  XXII 

Solve  the  following  equations  : 

1.  cos  x  —  V3  sin#=  1.  7.    cos  ex.  -f  sin  a  =  —  V2. 

2.  sin  x  —  V3  cos  #  =  1.  8.    sin  m0+  sin  nO  =  0. 

3.  sin  ^  +  V3  cos  ^  =  V2.  9.    cos  w^  +  cos  n^  =  0. 

4.  V3  sin  0  —  cos  ^  =  V2.  10.    3sin#  +  2cos^  =  2. 
5~    sin  6  +  cos  ^  =  V2.  11.    6  cos  0  —  3  sin  0  =  3. 
6.    cos  a  —  sin  a  =  -|-  V2-  12.    4  cos  #  —  3  sin  0  =  5. 


128  PLANE   TRIGONOMETRY 

~~  13.  sin  7  x  —  sin  4  x  +  sin  x  —  0. 

-  14.  sin  5  x  —  sin  3  x  +  sin  a;  =  0. 

15.  sin  1  x  —  sin  x  =  sin  3  a?. 

16.  sin  4x—  sin  2  x  =  cos  3  #. 

17.  cos  6  +  cos  2  0  +  cos  3  0  =  0. 

18.  sin  0  +  sin  2  0  +  sin  3  6  =  0. 

19.  cos  7  0  —  cos  0  =  —  sin  4  0. 

20.  cos  2  0  -  cos  0  -  sin  2  0  +  sin  0  =  0. 

21.  sin  4  0  —  sin  3  0  +  sin  2  0  —  sin  0  =  0. 

22.  cos70  +  cos50+eos30+cos0  =  0. 

23.  2  cos  2  0  =  cos  3  0  +  sin  0. 

24.  cos  &<^  —  cos  (k  —  2)  0  =  sin  0. 

25.  sin  5  0  cos  0  —  sin  6  0  cos  2  0  =  0. 


26. 


27.  COS  3  0  +  2  COS  0  =  0. 

28.  cos20-f-sin30  =  0. 

29.  cos  5  0  +  cos  0  =  V2  cos  3  0. 

30.  sin  0  +  V3  cos  4  0  =  sin  7  0. 

31.  cos20-cos20  =  0.  36.  cot  0- tan 0=2. 

32.  cos  3  0  +  8  cos30  =  0.  37.  sec  0  —  esc  0  =  2  V2. 

33.  sin  3  0  -  8  sin30  =  0.  38.  cot  2  0- cot  0  =  -  2. 

34.  sin20  +  3sin0  =  0.  39.  sec  4  0-  sec  20=  2. 

35.  esc  0  —  cot  0  =  V3.  40.  sec  0  +  esc  0  =  2  V2. 

41.  tan  3  0  +  tan  2  0  +  tan  0=0. 

42.  tan  30- tan  20 -tan  0=0. 

43.  tan  3  0  +  tan  0  =  2  tan  2  0. 

44.  sin  5  0  cos  0  -  sin  4  0  cos  2  0  =  0. 

vr 


GENERAL  SOLUTION  OF  TRIGONOMETRIC  EQUATIONS     129 


92.   Changes  in  sign  and  magnitude  of  the  expression  a  cos  a? 

4-  &  sin  x.     In  connection  with  the  solution  of  equations  of  the 

form  .   ,    . 

a  cos  x  4-  o  sin  x  =  U, 

it  is  often  useful  to  trace  the  changes  in  sign  and  magnitude 
of  the  left  member  of  the  equation  as  x  increases  from  0° 
to  360°. 

The  simplest  case  occurs  when  a  —  I  and  6  =  1;  in  which 
case  we  have  simply  sin  x  4-  cos  x  to  examine.  Proceeding  as 
in  Art.  90  we  have 

cos  x  4-  sin  x  =  V2  —  sin  x  4 cos  x 

LV2  V2         J 

=  V2(sin  x  cos  45°  4-  cos  x  sin  45°) 
=  V2sin<>4-450). 

For  convenience  we  replace  cos  x  4-  sin  x  by  y,  and  then,  form- 
ing the  equation  y  =  V2  sin  (x  +  45°),  we  form  the  following 
table  of  values. 

Plotting  the  graph  by  the  method  explained  in  Art.  48,  we 
have  the  following  result. 


X 

y 

0° 

1 

45° 

V2 

90° 

1 

135° 

0 

180° 

-1 

225° 

-V2 

270° 

-1 

315° 

0 

360° 

1 

0 


/T\ 


273 


~Kd 


Since  the  greatest  value  that  the  sine  of  any  angle  can 
have  is  1,  the  maximum  value  of  this  expression  occurs  when 
sin  (x  4-  45°)  =  1,  i.e.  when  z  =  45°.  This  gives  V2  as  the 
maximum  value  of  the  expression  sin  x  4-  cos  x. 

In  like  manner,  the  minimum  value  of    the  expression   is 
-  V2,  which  corresponds  to  the  angle  x  =  225°. 
COKANT'S  TRIG.  —  9 


130  PLANE   TRIGONOMETRY 

If  the  table  of  values  is  extended,  and  the  graph  is  plotted  for 
values  of  x  greater  than  360°,  the  values  of  ?/,  i.  e.  of  cos  x  +  sin  #, 
will  be  repeated  in  their  original  order ;  that  is,  cos  x  +  sinx 
is  a  periodic  function  with  a  period  of  360°.  (See  Art.  49,  p.  71.) 

93.  When  a  or  5,  or  both  a  and  £,  are  different  from  unity, 
the  process  is  slightly  modified,  as  follows  : 

a  cos  x+  b  sin  x  =  Va2  +  b2l — ^      -  cos  x  -\ —      -  sin  x  ] 

=  Va2  -f-  b2  (cos  x  cos  a  +  sin  x  sin  a) 
=  Va2  -f-  b2  cos  (x  —  a) . 

Here,  as  is  readily  seen  from  the  figure  on  the 
b  margin  of  the  page,  it  has  been  assumed  that  a  is 

the  angle  whose  cosine  is  —   a         and  whose  sine  is 
When  a  and  b  are  known,  a  can  be  found,  as  in 


•Va2  +  b2 

Art.  90,  p.  124. 

The  table  of  values  can  then  be  obtained  and  the  graph  con- 
structed, as  in  the  preceding  case. 

Since  cos  (x  —  a)  has  1  for  its  maximum  value  and  —  1  for  its 
minimum  value,  the  expression  a  cos  x  +  b  sin  x  has  Va2  +  b2 
for  its  maximum  value  and  —  Va2  —  b2  for  its  minimum  value. 

NOTE.  In  computing  the  table  of  values  for  the  purpose  of  constructing 
the  graph,  the  values  of  y  can  always  be  obtained  directly  from  the  expres- 
sion as  it  is  originally  given,  without  any  reduction  whatever.  This  is 
sometimes  preferable;  and  in  certain  cases,  as  for  example  the  functions 
given  in  Examples  7,  9,  10,  and  11  in  the  following  set,  it  is  easier  to  com- 
pute the  values  directly  than  to  compute  them  after  transforming  the 
expression. 

EXERCISE  XXIII 

Trace  the  changes  in  sign  and  magnitude  of  the  following 
expressions  as  x  increases  from  0°  to  360°.  Find  the  period 
and  construct  the  graph  in  each  case. 

1.    sin  x  —  cos  x.  5.    sin  x  +  V3  cos  x.       9>    cos  3  9. 

10'    Sm  8  0' 


2.  V3sinz  +  cosz.     6.    2  sin  x  +  3  cos  x. 

11.  tan  20. 

3.  sin*  +  V3cos*.      7.    cos  20.  ^  sin  20-  sin* 

4.  V3  sin  x  —  cos  x.     8.    sin  6  cos  6.  cos  2  0  +  cos  6 


CHAPTER   XII 
THE  OBLIQUE  TRIANGLE 

94.  The  law  of  sines.  Let  A,  B,  0  denote  the  angles  of  a 
triangle,  and  a,  b,  c  respectively  the  sides  opposite. 

From  any  vertex,  as  (7,  draw  CD  perpendicular  to  AB,  meet- 
ing AB,  or  AB  produced,  in  D. 


A  D  B  A  B 

From  the  first  figure  we  have 


Also, 


=  b  sin  A. 


-  =  sm  B. 
a 

.•.  h  =  a  sin  B. 


Equating  these  values  of  h  we  have 

b  sin  A  =  a  sin  B. 

From  the  second  figure  we  have 


-  =  sm  A. 
b 


=    sn 


A. 


Also, 
whence  as  before, 


-  =  si 

b  sin  A  =  a  sin 
131 


132  PLANE   TRIGONOMETRY 

Therefore  in  either  case  we  have  the  same  result, 
b  sin  ^4.=  a  sin  J9; 
a  -  sin  A 


In  like  manner  drawing  perpendiculars  from  the  vertices  A 
and  B  to  the  opposite  sides  respectively  we  can  prove  that 

b  _  sin 
G     sin 

and 


c      sm  0 

The  results  obtained  in  (1),  (2),  and  (3)  enable  us  to  state 
the  law  of  sines  as  follows : 

The  sides  of  a  triangle  are  proportional  to  the  sines  of  the 
opposite  angles. 

Equations  (1),  (2),  and  (3)  are  often  combined  and  written 
in  the  following  manner  : 


sin  A      sin  B     sin  0 

95.  The  geometric  meaning  of  each  of  the  three  ratios  just 
stated  will  be  understood  from  the  following : 

Let  ABO  be  any  triangle,  and  let  a  circle  be  circumscribed 
about  the  triangle.  From  the  center  0  to  the  vertices  of  the 
triangle  draw  the  radii  OA,  OB,  00,  respectively,  and  also 

draw  OD  perpendicular  to  AB. 
By  geometry 


From  this  we  have 


=  r  sin  C. 
.*.  c  =  2  r  sin  C. 
In  like  manner  it  can  be  proved  that 

a=  2rsin  A* 
and  b  =  2  r  sin  B. 


. 


THE  OBLIQUE  TRIANGLE 


133 


Equating  the  values  of  2  r  obtained  from  these  three  equa- 
tions we  have  a  i  c 

2r  =  - — -  =  -r—  =  -^-~ -    That  is, 
Bin  .4     sin  If     sin  O 

The  ratio  of  any  side  of  a  triangle  to  the  sine  of  the  opposite 
angle  is  equal  to  the  diameter  of  the  circumscribed  circle. 

96.  The  law  of  cosines.  Let  ABO  be  any  triangle,  and  let 
(7Z),  the  perpendicular  from  the  vertex  0  to  the  opposite  side, 
meet  AB,  produced  if  necessary,  in  D. 


D 


B      A 


From  the  first  figure  we  have 


=  52  +  ^2  _  2  c  -  b  cos  A. 


2  be 

From  the  second  figure  we  have 
«2  =  h*  +  BD* 
=  h?  +  (AD  -  c) 
=  £2  +  AD*-2c 
=  b2  +  c2  —  2  c  -  b  cos  A. 


2  be 


Therefore,  the  same  result  is  obtained  for  both  triangles. 
In  like  manner,  drawing  perpendiculars  from  A  and  B  to  the 
opposite  sides  respectively,  we  can  prove  that 


and 


cos  B  = 


cos  C'  = 


(2) 
(3) 


134  PLANE   TRIGONOMETRY 

Equations  (1),  (2),  and  (3)  are  often  useful  when  expressed 
in  the  following  form  : 


(4) 

C. 

The  law  of  cosines  can  now  be  stated  as  follows  : 

The  square  of  any  side  of  a  triangle  is  equal  to  the  sum  of  the 

squares  of  the  other  two  sides  minus  twice  their  product  into  the 

cosine  of  the  included  angle. 

yv 

97.   The  law  of  tangents.     We  have  already  proved  that,  in 

.  .       i  a     sin  A 

any  triangle,  -  =  —  • 

Therefore,   considering  this  equation  as  a  proportion,  and 
taking  the  four  quantities  by  division  and  composition, 

a  —  b__  sin  A  —  sin  B 
a  +  b      sin  A  +  sin  B 


2  sin  *±**  cos  ^p? 

~  L 

cot^i^tanAzi^. 


a-b  2 


2 

In  like  manner  it  can  be  proved  that 

A-C 

tan  — 


and 


THE   OBLIQUE   TRIANGLE  135 

The  law  of  tangents  can  now  be  stated  as  follows  : 

The  difference  of  two  sides  of  a  triangle  is  to  their  sum  as  the 
tangent  of  half  the  difference  of  the  opposite  angles  is  to  the  tan- 
gent  of  half  their  sum. 

NOTE.  In  using  the  formulas  of  this  section  it  is  better  to  let  the  greater 
side  and  the  greater  angle  precede  the  smaller  in  all  cases.  The  formulas 
are  true,  whichever  order  is  used  ;  but  if  the  smaller  side  and  the  smaller 
angle  precede  the  greater  side  and  the  greater  angle  respectively,  negative 
numbers  are  introduced,  and  if  logarithms  are  to  be  employed,  these  num- 
bers should  be  avoided  whenever  it  is  possible  to  do  so. 

98.  The  given  parts.     In  the  solution  of  oblique  plane  tri- 
angles four  cases  occur.     In  each  case  three  parts  are  given,  as 
follows  : 

1.  One  side  and  two  angles. 

2.  Two  sides  and  the  angle  opposite  one  of  them. 

3.  Two  sides  and  the  included  angle. 

4.  Three  sides. 

The  formulas  developed  in  Arts.  94-97  are  sufficient  for  the 
solution  of  every  possible  case  that  can  arise.  These  cases  will 
now  be  considered  separately. 

99.  CASE  1.     Given  one  side  and  two  angles.     Let  the  given 
angles  be  A  and  J5,  and  the  given  side  a.     The  formulas  for 
solution  are  as  follows  : 


b  _  sin  B  7  _  a  sin  B 

2.         —  -    —  -,  .    .    0   -          ;  — 

a      sin  A. 


c      sin 


> 
a      sin  JL'  sin  A 

Ex.  i.    Given  a  =  467,  A  =  56°  28',  B  =  69°  14';  find  the  re 
maining-  parts. 

The  work  may  be  conveniently  arranged  as  follows  : 
C  =  180°  -  (.4  +  B)  =  54°  18'. 

(1)  By  natural  functions. 

b  =  a  x  sin  B  -  sin  A  =  467  x  0.9350  t  0.8336  =  523.8. 
c  =  a  x  sin  C  •*-  sin  .4  =  467  x  0.8121  -*•  0.8336  =  454.95. 


\\ 


136  PLANE   TRIGONOMETRY 

(2)  By  logarithms. 

log  b  =  log  a  +  log  sin  B  —  log  sin  A 

=  log  «  +  log  sin  B  +  colog  sin  A. 
log  c  =  log  a  +  log  sin  C  —  log  sin  A 

=  log  a  +  log  sin  C  +  colog  sin  A. 

log  a  =  2.66932  log  a  =  2.66932 

log  sin  B  =  9.97083  -  10  log  sin  C  =  9.90960  -  10 

colog  sin  A  =  0.07906  colog  sin  A  =  0.07906 

2.71921  2.65798 

b  =  523.85  c  =  454.97 

NOTE.  To  insure  accuracy  the  student  should  check  all  results  by  solving 
each  problem  by  a  second  method,  entirely  independent  of  the  first  ;  or  by 
the  same  method,  using  a  different  combination  of  parts.  In  the  case  under 
consideration  it  is  usually  sufficient  to  employ  the  same  method,  i.e.  the  law 
of  sines,  combining  the  parts  in  a  manner  different  from  that  employed  in 
the  first  place.  For  example,  after  c  has  been  found  we  can  solve  again  for 

b  by  the  formula  b  =  csm/f  ,  as  follows  : 
sin  C 

log  c  =  2.65798 
log  sin  B  =  9.97083  -  10 
colog  sin  C  =  0.09040 

log  b  =  2.71921         b  =  523.85  check. 

EXERCISE  XXIV 
Solve  the  following  triangles  : 

1.  Given  a  =  438.3,   A  =  43°  50'  24",   B=  69°  30' 
An*.    C=  66°  39'  24",    b  =  592.74,   c  =  580.*. 

•1*1 

2.  Given  b  =  421,   A  =  31°  12',   B  =  36°  20'. 

Ans.    (7=112°  28',   a  =  368.08,   c  =  656.63. 

{  3.   Given  a  =  412,   4  =  58°U',   B  =  65°  37'. 
Ans.    0=  56°  9',   5  =  441.37,    c  =  402.45. 

4.    Given  6  =  81.5,    B  =  43°  44'  18",    0=  75°  2'  42". 
4  =61°  13',   a  =103.32,    c=  113.89. 


5.  Given  c  =  77.93,    B  =  22°  15'  20",    O=  41°  50'  30". 
Aw«.  A  =  115°  5&  10",   a  =  105.07,   5  =  44.23. 

6.  Given  c  =  6.98,   A  =  25°  7'  10",    (7=  36°  12'  24". 
Ans.  B  =  118°  40'  £$",   4  =  5.016,    b  =  10.37. 


THE   OBLIQUE   TRIANGLE  137 

7.  Given  a  =  928.4,   A  =  61°  17'  15",    6V=  58°  18'  40". 

Am.  B  =  60°  24'  5",   c  =  900.78,    ft  =  920.45. 

8.  Given  a  =  328.4,    A  =  29°  41'  12",   B  =  37°  50'  24". 
Ana.   C  =11 2°  28'  24",    5  =  406.77,    c  =  612.73. 

9.  Given  A  =  64°  35',  0=  73°  49',  a  =  213.47. 
Ans.    B  =  41°  36',  5=156.92,  c=  226.98. 

10.  Given  ^1  =  41°  23'  47",  B  =  124°  49',  5  =  65.536. 
Am.     0=  13°  47'  13",  a  =  52.788,  c  =  19.023. 

11.  Two  points,  A  and  ^,  are  separated  by  a  body  of  water. 
In  order  to  find  the  distance  between  them  a  line  AQ  is  meas- 
ured 612.3  ft.  in  length,  and  the  angles  BAG,  ACB  are  meas- 
ured  and   are  found  to  be  49°  17'  and  68°  11'    respectively. 
What  is  the  distance  from  A  to  B  ? 

12.  It  is  desired  to  find  the  distance  of  a  lighthouse  A  to 
each   of   two   stations   B,   C,   situated   on   shore,    and   in    the 
same  horizontal  plane  with  the  base  of  the  lighthouse.     BC 
is  21  miles,  Z.ABO  is  39°  38',  and  ZACB  is  74°  56'.     Find  AB 
and  AC. 

13.  The  angles  of  elevation  of  a  balloon  that  has  ascended 
vertically  between  two  stations  one  mile  apart  on  a  level  plain, 
and  in  the  same  vertical  plane  with  the  balloon,  are  29°  41'  and 
37°  17'  respectively.     What  is  the  distance  of  the  balloon  from 
each  station,  and  what  is  its  vertical  height  above  the  plain  ? 

14.  Solve    the   preceding   problem    on  the  supposition  that 
both  the  stations  are  on  the  same  side  of  the  balloon. 

15.  To  find  the  width  of  a  stream  a  line  AB,  351  ft.  long,  is 
measured  on  one  side,  parallel  to  the  bank.     On  the  opposite 
side  of  the  stream  a  stake  C  is  set,  and  the  angles  CAB,  CBA, 
are  observed  and  are  found  to  be  38°  17'  and  31°  29'  respec- 
tively.    What  is  the  width  of  the  stream  ? 

16.  From  the  top  and  bottom  of  a  column    the    angles   of 
elevation  of  the  top  of  a  tower  236  ft.  high  standing  on  the 
same   horizontal    plane   are  44°  27'  and  61°  31'  respectively. 
What  is  the  height  of  the  column  ? 


138  PLANE   TRIGONOMETRY 

17.  When  the  altitude  of  the  sun  is  49°  52',  a  pole  standing 
on  the  slope  of  a  hill  inclined  16°  55'  to  the  level  of  the  plain 
casts  a  shadow  directly  down  the  hill  a  distance  of  45  ft.  8  in. 
What  is  the  height  of  the  pole  ? 

18.  An  observer  in  a  balloon  measures  the  angle  of  depres- 
sion of  an  object  on  the  ground  and  finds  it  to  be  63°  58'.    After 
ascending  vertically  582  ft.  he  finds  the  angle  of  depression  of 
the  same  object  74°  49'.     What  was  the  height  of  the  balloon 
at  the  time  of  the  first  observation  ? 

19.  From  a  ship  the  bearings  of  two  objects  were  found  to 
be  N.N.W.  and  N.E.  by  N.,  respectively.     After  sailing  due 
east  10  miles  the  two  objects  were  in  a  line  bearing  W.N.W. 
How  far  apart  were  the  objects  ? 

NOTE.  For  an  explanation  of  the  term  "bearing,"  and  for  instruction  in 
reading  angles  by  means  of  the  compass,  see  p.  176. 

20.  From  a  ship  a  lighthouse  bears  N.  21°  12'  E.     After  the 
ship  has  sailed  S.  25°  12'  E.  2|  miles  the  lighthouse  bears  due 
north.     Find  the  distance  of  the  lighthouse  from  the  last  point 
of  observation. 

100.  CASE  2.  Given  two  sides  and  the  angle  opposite  one  of 
them.  Let  the  given  parts  be  the  sides  a  and  &,  and  the  angle 
A.  The  required  parts  can  be  found  in  the  following  manner  : 

By  the  law  of  sines 

(1) 


sin  A      a  a 

From  this  equation  the  angle  B  can  be  found. 

Then,  C=  180°  -  (4  +  B). 

Also,  ^  =  ^4,        .'.c  =  ^™?.  (2) 

a      sin  A  sin  A 

In  solving  for  the  angle  opposite  the  second  side,  in  this 
case  the  angle  B,  it  is  to  be  noted  that  two  solutions  are  pos- 
sible, since  the  sines  of  supplementary  angles  are  equal  (Art. 
53,  p.  79). 

The  following  considerations  will  determine  the  number  of 
solutions  for  any  given  set  of  conditions. 


THE   OBLIQUE   TRIANGLE 


139 


If  a  >  b,  then  A  >  B,  and  B  is  necessarily  an  acute  angle, 
since  a  triangle  can  have  but  one  obtuse  angle.  Therefore 
there  is  one  and  only  one  solution. 

If  a  =  b,  then  A  =  B,  and  both 
A  and  B  are  acute  angles.  There- 
fore there  is  one  and  only  one  solu- 
tion, an  isosceles  triangle. 

If  a  <  b,  then  A  <  B,  and  A  is  an  FlG- 1- 

acute  angle.     In  this  case  B  may  One  solution,  a> 6 

be  either  acute  or  obtuse,  and  there  will  be  two  solutions  if 
a  >  CD,  the  perpendicular  drawn  from  the  vertex  C  to  AB, 
produced  if  necessary.  That  is,  either  of  the  two  triangles 
ABl  C,  AB2  C,  will  satisfy  the  given  conditions.  But  the  perpen- 
dicular CD  =  b  sin  A.-  Therefore,  if  A  is  acute  and  #<&,  and 

c 
c 


b  sin  A 


FIG.  2. 
Two  solutions,  a  >  b  sin  A 


FIG.  3. 

One  solution,  a  =  b  sin  A 


if  a  >  b  sin  A,   there   are    two    solutions.     The    angles 
AB^C,    are    supplementary,    since    /.AB1C=/.B1B^C.     Both 
angles  are  given  by  the  formula 


If  a  =  b  sin  A,  that  is,  if  a  is  equal  to  the  perpendicular  CD, 
there  is  but  one  solution,  a  right  triangle.  This  is  also  seen  from 
the  fact  that  when  a=  b  sin  A,  the  value  of  sin  B  reduces  to 

unity.     This  gives  B  =  90°. 

If  a  <  b  sin  A,  that  is,  if  a  is  less 
than  the  perpendicular  CD,  there  is 
no  solution,  and  the  triangle  is  impos- 
sible. This  is  also  seen  from  the  fact 
that  when  a<bsinA,  the  fraction 
FIG.  4.  b  sin  A  is  ter  than  unit  But 

No  solution,  a  <  6  sin  A  a 


140  PLANE   TRIGONOMETRY 

this  fraction  is  in  all  cases  equal  to  sin  B ;  and  as  the  sine  of  an 
angle  can  never  exceed  unity  the  triangle  is  therefore  impossible. 

These  results  may  be  summarized  as  follows : 

Two  solutions. 

A  acute,  a  <  6,  and  a  >  b  sin  A. 

One  solution. 

(0)    A  obtuse  and  a  >  b. 

(5)   A  acute  and  a  =  b  sin  A. 

(c?)    A  acute  and  a  >  b. 

No  solution. 

(a)  A  acute  and  a  <  b  sin  A. 

(b)  A  obtuse  and  a  =  b  or  a  <  b. 

To  determine  the  number  of  solutions,  first  note  whether  A 
is  acute  or  obtuse.  Then,  on  examining  the  different  cases  just 
studied,  it  is  seen  that  there  can  never  be  more  than  one  solu- 
tion unless  A  is  acute  and  the  Me  opposite  A  is  less  than  the  side 
adjacent.  In  this  case  there  may  be  two  solutions,  one  solution, 
or  no  solution. 

The  comparison  between  a  and  b  sin  A  is  often  most  con- 
veniently made  by  means  of  the  natural  value  of  sin  A.  In 
many  cases  the  computation  can  be  performed  mentally  ;  for 
all  that  is  now  desired  is  to  determine  whether  a  is  less  than, 
equal  to,  or  greater  than  b  sin  A. 

If  logarithms  are  used,  we  compute  log  sin  J5.  The  results 
are  as  follows. 

(a)  log  sin  1?>0,  no  solution. 

(b)  log  sin  B  =  0,  one  solution,  a  right  triangle. 

(V)  log  sin  B  <  0,  one  solution  if  a  >  5,  and  two  solutions  if 
a<  b  and  A  is  acute. 

The  student  should  bear  in  mind  that  the  given  parts  are 
not  necessarily  a,  b,  and  A  ;  they,  may  be  any  two  sides  and 
the  angle  opposite  one  of  them.  If  other  parts  are  given  than 
those  mentioned  above,  the  proper  modifications  should  be 
made  in  the  formulas  for  determining  the  number  of  solutions. 

Ex.  1.  Given  a  =  26,  b  =  72,  A  =  30°  ;  find  the  remaining 
parts. 

Since  sin  A  —  \,  we  have  b  sin  A  =  36.  Hence,  the  triangle  is  impossible 
as  a  <  36. 


THE   OBLIQUE   TRIANGLE  141 

Ex.  2.    Given  a  =  88,  b  =  103,  A  =  120°;  find  the  remaining 
parts. 

Here  A  is  obtuse  and  a  <  b  ;  therefore  the  triangle  is  impossible. 

Ex.  3.    Given  a  =738.4,  6  =  1185.7,.  ^  =  79°  38'    40";  find 
the  remaining  parts. 

Solving  by  logarithms  we  proceed  as  follows  : 


a 

logb  =    3.07397 
log  sin  A  =    9.99287  -  10 

colog  a  ="7.13171-10  Since   log  sin5>°>  there  is  no 

log  sin  5  =  10.19855  -10 


Ex.  4.    Given  a  =  28.2,  e  =  45.65,  A  =  38°  1'  7.5"  ;  find  the 
remaining  parts. 

Proceeding  as  in  Ex.  3  we  have  , 


a 

logc  =    1.65944  ...  C=  90°,  and  the  triangle  is  a 

log  sin  A  =    9.79081  -  10  right  triangle. 

colog  a  =    8.54975  -  10 
log  sin  C  -  10.00000  -  10 

Solving  for  B  and  b  by  the  usual  methods  employed  in  the  case  of  right 
triangles  (Arts.  26  and  27,  pp.  36-38),  we  find  B  =  51°  50'  52.5",  b=  35.998. 

Ex.  5.  Given  a  =  67.53,  b  =  56.82,  A  =  77°  14'  19"  ;  find  the 
remaining  parts. 

Here  a  >  b  and  A  is  acute;  therefore  there  is  but  one  solution. 
The  unknown  parts  are  found  in  the  following  manner  : 

log  b  =  1.75450 
log  sin  A  =  9.98914  -  10 

colog  a  =  8.17050  -10  C  =  180°-(A+B) 

__  4gO  00  1    KAff 

log  sin  B  =  9.91414-  10 

•••*=55°8'47"'  Check: 

log  b  =  1.75450  log  «  =  1-82950 

log  sin  C  =  9.86843  -  10  log  sin  C  =  9.86843  -  10 

colog  sin  A  =  0.08586  colog  sin  A  =  0.01086 

log  c  =  1.70879  log  c  =  1.70879 

.-.  c=  51.143.  c  =  51.143 


142  PLANE   TRIGONOMETRY 

Ex.6.    Given   «=  168.32,  5=221.46,  4  =  33°  39' 16";    tind 
the  remaining  parts. 

In  this  case  the  simplest  method  of  finding  the  number  of  solutions  is  to 
obtain  the  value  of  b  sin  A  by  multiplying  the  value  of  b,  221.46,  by  the 
natural  value  of  sin  A,  and  comparing  the  result  with  168.32,  the  value  of  a. 
The  sine  of  33°  39'  16"  is  approximately  0.55.  Hence,  it  is  seen  at  a  glance 
that  b  sin  A  is  a  trifle  over  one  half  of  221.46;  that  is,  much  less  than  a 
Hence,  since  A  is  acute  and  a  <  &,  there  are  two  solutions. 

The  work  of  computation,  exhibited  in  compact  form,  is  as  follows : 


log  b  =  2.34529  log  a  =  2.22613 

log  sin  A  =  ,9.74365  -  10  log  sin  C  =  9.99396  -  10 

colog  a  =  7.77387  -  10  colog  sin  A  =  0.25635 


2.22613 
9.35729  -  10 
0.25635 


log  sin  B  =  9.86281  -  10  log  c  =  2.47644  1.83977 

.-.  Bl  =  46°  48'  50",  .-.  cx  =  299.53,     c2  =  69.147. 

B2  =  133°  11'  10". 
.-.  C  =  99°  31'  54",  or,  13°  9'  34". 

NOTE.  The  method  of  checking  results  is  the  same  as  that  used  in  con- 
nection with  Case  1.  In  Ex.  5  above  the  check'  work  for  c  is  given.  After 
a  little  practice  this  work  can  be  performed  with  great  rapidity.  Every 
result  obtained  by  the  student  should,  be  subjected  to  a  check  of  some  kind. 

• 
EXERCISE   XXV 

1.  Determine  the  number  of  solutions  in  each  of  the  follow- 
ing cases : 


(1) 

a  =  30, 

5  =  60, 

4  =  30°. 

(2) 

a  =  20, 

5  =  60, 

4  =  30°. 

(3) 

«  =  40, 

5=<;o, 

4=30°. 

(4) 

a  =  750, 

5  =  638, 

A  =  69°  30'. 

(5) 

a  =  38.  8, 

5  =  45.5, 

4  =  60°. 

(6) 

a  =  226, 

5  =  196, 

4  =  123°  40'. 

2.    Given 

a=l  09.68, 

e  =  467, 

A=  13°  35'; 

find 

(7=90', 

^  =  76°  25', 

5  =  453.94. 

3.    Given 

a  =392, 

5  =  124, 

A  =  36°  41'  42"; 

find 

.5=10°  53'  45" 

<7=132°24'33" 

£  =  484.37. 

4.    Given 

a  =  168.2, 

5  =  218.6, 

4  =  34°22;50"; 

\f  r~" 

find     . 

g1  =  47°12'49", 

6\  =  98°24'21", 

e1  =  294.67. 

5o=132°47'll", 

(79  =12°  49'  59", 

<?9  =  6(:>.16. 

THE   OBLIQUE   TRIANGLE  143 

5.  Given  6  =  8472.2,              c  =  3211.7,  (7=16°  33' 45"; 
find    ^  =  114°  40'  42",  ^  =  48°  45'  33",  «1  =  10238. 

^2  =  320  11' 48",     B2=UI°U' 21",  «2=6003.4 

6.  Given  a  =  506,  6  =  432,  ^  =  36°7'12"; 
find     ,6  =  30°  13',             67=113°  39' 48",      c=  7-86.22. 

7.  Given  a  =  36.27,                6  =  23.96,  5=30°  26'  14"; 
find    ^41  =  50C4'24",        ^  =  99°  29' 22",  ^  =  46.65, 

A2  =129°  55'  36",  (72=19°38'10",  ca=l 


' 


8.    Given  «  =  283.4,  5  =  268.5,  JL  =  60°  40' 26"; 

find     J5=55°41/23",      (7=  63°  38' 11,         c=  291.25. 


9.    Given  a  =  158,  6  =  179,  J.  =  21°17' 22"; 

find    ^  =  24°  17' 18",  6\  =  134°25' 20",  ^=310.8, 

52  =  15lfp.42'42",  <?2  =  2°  59'  56",  «2  =  22.767. 

10.  Given   a  =  628. 2,  6  =  234.4,  4  =  119° 40'  40"; 
find      ^=18° 54' 58",  (7=41°  24' 22",        ^=478.22. 

11.  Given    a  =  86. 14,               6  =  97.82,  ^  =  38°  15' 14"; 
find    ^  =  44°  40' 42",  C\  =  97°4'4",  c?1=138.07, 

672  =  6°25'28",  ^2  =  15.57. 


12.  Given   a  =  158,  6  =  179,  ^  =  21°  17' 22"; 
find      j5=24°17'18",  0=  134°  25'  20",      c=  310.8, 

5' =  155° 42' 42",  0'  =  2°  59'  56",  c'  =  22.77. 

13.  Given   a  =  36. 38,  6  =  23.92,  A  =  39°  2'  14"; 
find      J5=24°27'49",  0=  116°  29'  57",      <?  =  51.69. 

14.  Given    a  =  0.09593,  6  =  0.16864,  5=125°  33'; 
find     ^1=27°  34'  12",  0=  26°  52'  48",        e=  0.09375. 

15.  Given   a  =  354.16,  6  =  433.86,  .A  =  36°1'4"; 
find    ^  =  46°  5' 5",  ^  =  97°  53' 51",  ^  =  596.57, 

R2  =133°  54'  55",  O2  =  10°  4'  1",  ^2=105.26. 


144  PLANE   TRIGONOMETRY 

16.  Given  a  =  25.675,  6  =  50.139,  £  =  68°  4'  14"; 
find      4=28°  21'  42",     C=83°34'4",          e=53.709. 

17.  Given  a=542.99,  6  =  310.71,  ^=122°  49'  17"; 
find      5=28°  44'  34",     (7=  28°  26'  9",          e=307.66. 

18.  Given    a=  346.66,  <?=412.33,  J.=  24°19'  51"  ; 
find     ^  =  126°  19'  31",  ^  =  29°  20'  38",       ^  =  677.87, 

£2  =  5°0'47",        <72=150°39'22",     62=  73.524. 

19.  Given   a  =  56.82,  6  =  67.53,  ^=77°  14'  19"; 

find     ^L  =  55°8'47",       (7=47°  36'  54",       c  =  51.14. 

101.   Given  two  sides  and  their  included  angle. 

First  method.  When  one  angle  O  is  given,  the  remaining 
angles  can  be  found  by  the  law  of  tangents  (Art.  97,  p.  134), 
which  can  be  expressed  in  the  following  manner  : 


2          a  +  b  2 

The  angle  —  -  —  -  =  90°  —  —  •     Hence,  its  value  is  known,  and 

'—  "2 

the  value  of  —  -  —  can  be  obtained  from  the  above  equation. 

2 

The  values  of  A  and  B  can  then  be  found  as  follows  : 

A+B  ,  A-B_ 

~~~       ~~ 


,nd 


The  remaining  side  c  can  now  be  found  by  the  law  of  sines 
in  either  of  the  two  following  ways : 

a  sin  C  b  sin  0 

c  =  — — — ,  or  c  =  — : — —  • 
sin  A.  sin  ±f 

/Second  method.      The  third  side  c  can  be  found  directly  by 
the  law  of  cosines  (Art.  96,  p.  133),  as  follows  : 


THE   OBLIQUE   TRIANGLE  145 

and  the  angles  A  and  B  can  then  be  found  by  the  law  of  sines, 

as  follows :  .    n  -L    -    n 

•     A  —  a  sm          '     7?  —     Sln 
c  c 

Third  method.     In  the  triangle  ABC  let  the  given  parts  be 
a,  0,  C.     From  the  vertex  B  draw  BD  perpendicular  to  AC. 

Then,'  BD  =  a  sin  (7, 

and  1)0=  a  cos  (7. 

Now 


Substituting  in  this  equation  the  values  of  BD  and  D C,  we 
have 


0  —  a  cos  (7 

In  like  manner,  drawing  a  perpendicular  from  A  to  the  side 
BO  it  can  be  proved  that 

5  sin  O 


tan 


cos  C 


The  third  side  can  now  be  found  by  the  law  of  sines,  as 
under  the  first  method. 

NOTE.  The  first  method  is  the  best  for  use  when  all  the  unknown  parts 
are  desired.  If  only  the  third  side  is  desired,  the  second  method  can  be 
used  to  advantage.  The  second  and  third  methods  are  not  suitable  for 
computation  by  means  of  logarithms. 

Ex.  1.  Given  a=  138.65,  0  =  226.19,  (7=59°  12'  54";  find 
the  remaining  parts. 

b-a=   7a,54  log(b-a)=    1.94221 

B:t:!f^rv>  ^.^^±4=10^46-10 

B  +  A  =    60o  23/33"  colog(6  +  a)=    7.43790  -  10 

Q  •  75  A 

n  "   A  log  tan  — — —  —    9.62557  —  10 

**~A  =   22°  53' 31"  2 

2  A=    37°  30' 2"  ^r''~-    22°58'31" 

B=    83°  17' 4" 
CONANT'S  TRIG. — 10 


146  PLANE   TRIGONOMETRY 

Check: 

loga=    2.14192  log  b  =  2.35447 

log  sin  C  =    9.93494  -  10  .  log  sin  C  =  9.93494  -  10 

colog  sin  A  =  10.21554  -  10  colog  sin  B  -  0.00299 

logc=    2.29240  log  c  =  2.29240 

c=    196.06  c=    196.06 

NOTE.  In  the  solution  of  this  problem  b  precedes  a  since  b  >  a.  (Compare 
Art.  97,  p.  134.)  In  finding  c  we  use  A  rather  than  B,  because  B  is  so  near 
90°  that  any  solution  obtained  by  means  of  its  sine  is  likely  to  be  inaccurate. 

NOTE.  In  Ex.  1  the  check  solution  gives  a  result  exactly  equal  to  that 
obtained  by  the  original  solution.  In  the  work  near  the  top  of  p.  136  the 
check  solution  also  gave  a  result  exactly  equal  to  that  obtained  in  the  origi- 
nal solution.  In  general,  however,  the  check  solution  may  be  expected  to 
differ  slightly  from  the  original. 

Ex.  2.    Given  a  =  7,  c  =  9,  B  =  60°  ;  find  the  third  side  6. 

In  this  problem  the  second  method  furnishes  the  solution  with  the 
smallest  amount  of  labor. 

fe2  =  a2  +  c2  —  2  etc  cos  B, 

b  =  V49  +  81  -  2  •  .7  •  9  •  £  =  VtJ7. 
.-.  b  =  8.1854. 

EXERCISE  XXVI 

1.  Given   a  =  426,  6  =  582,  0=  52°  18'; 
find      A  =  46°  21'  16",     ^=81°  20' 44",       c  =  465.8. 

2.  Given   6  =  123,  c  =  211,  4  =  115°  22'; 
find      ^  =  41°  46' 45",      0=  22°  51'  15",      a  =  286.16. 

3.  Given  a  =  121. 6,  c  =  192.2,  B  =114° .42'; 
find      ^=24°  26' 49",      0=  40°  51'  11",      6  =  266.94. 

4.  Given  a  =  619,  6  =  515,  6^=39°  17'; 
find      A  =  84°  46'  10",     B=  55°  56'  50",       c=  393.56. 


< 


5.  Given   6  =  35.218,  c  =  54.176,  A  =  32° 48'  14"; 
find      ^=37°  49' 35",  0=  109°  22'  11",  a  =31.112. 

6.  Given   a  =  46.792,  c  =  61.234,  ^=45°  29' 16"; 

find      ^  =  49°  34' 5",  0=  84°  56'  39",  6  =  43.836. 


THE   OBLIQUE   TRIANGLE  147 

7.  Given   b  =  718.01,             c  =  228.88,  A  =  68°  55'  2"; 
find      B  =  92°  30'  47",      (7=  18°  3-1'  11",  a  =  670.61. 

8.  Given   5  =  2478.1,  c  =  5134.8,  A  =  137°  8'  49"; 
find      5  =13°  37'  43.  5",  0=  29°  13'  27.  5",  a  =7152.  5. 

9.  Given  a  =  4.1203,             5  =  4.9538,  O=  65°  38'  52"; 
find     A  =  -&4'  18",       B  =  65°  16'  50",  c  =  4.  9683. 


10.    Given   a  =  0.59217,  5  =  0.21833,          (7=  41°  37'  4"; 

find      ^1=119°  42'  18",   ^=18°  40'  38",      c  =  0.4528. 

11.  Two  objects  A  and  B  are  separated  by  a  body  of  water. 
In  order  to  find  the  distance  between  them  a  third  point  C  is 
chosen  from  which  each  of   these  points  is  visible,  and  the 
following  measurements  are  made:    CA  =  2560  ft.,  (7.5=3120 
ft.,  and  Z  ACB  =  105°  35'.     Find  the  distance  from  A  to  B. 

12.  If  two  sides  of  a  triangle  are   68.6   ft.   and   92.2   ft. 
respectively  and  the  included  angle  is  112°  42',  what  is  the 
third  side  ? 

13.  Find  the  distance  between  two  points  A,  B,  which  are 
separated  by  a  marsh,  when  the  distances  of  these  points  from 
a  third  point  C  are  4214  ft.  and  6932  ft.  respectively,  and  the 
angle  A  CB  is  51°  11. 

14.  In  an  isosceles  triangle  each  of  the  equal  sides  is  9  and 
the  included  angle  is  60°.     Find  the  third  side. 

15.  In  an  isosceles  triangle  each  of  the  equal  sides  is  9  and 
the  included  angle  is  120°.     Find  the  third  side. 

16.  There  are  two  points,  A,  B,  on  the  bank  of  a  river,  but 
owing  to  a  curve  in  its  course  it  is  impossible  to  measure  the 
distance  between  them   directly.     A  third  point  C  is  chosen 
such  that  the  distances  AC=l±6Q  ft.  and  5(7=1680  ft.  can 
be  measured,  and  the  angle  ACB  is  found  to  be  68°  42'  30". 
What  is  the  distance  from  A  to  B? 

17.  In  a  given  triangle  two  of  the  sides  are  6  and  9  respec- 
tively, and  the  included  angle  is  38°.     What  is  the  third  side? 

18.  The  diagonals  of  a  parallelogram  are  8  and  10  respec- 
tively, and  they  intersect  at  an  angle  of  60°.    What  are  the  sides 
of  the  parallelogram? 


148  PLANE   TRIGONOMETRY 

19.  If  two  sides  of  a  triangle  are  1468  and  2136  respectively 
and  the  included  angle  is  72°  21'  14",  what  are  the  values  of 
the  other  angles? 

20.  There  are  two  points,  A,  B,  so  situated  that  they  are  not 
visible  from  each  other,  and  there  is  no  other  point  from  which 
both  can  be  seen.     To  find  the  distance  from  A  to  B  two  other 
points  (7,  .Z),  are  selected  so  that  A  and  D  are  visible  from  (?, 
and  B  and  0  are  visible  from  D\  and  the  following  measure- 
ments are  made:    CD  =  826.5  ft.,  ZACD  =  121°  12',Z£OZ)  = 
58° 55',  ^ADC=  49°  12',  ^ADB  =  62°  38'.      What  is  the  dis- 
tance from  A  to  B? 

102.  Given  the  three  sides  a,  b,  c.  When  the  three  sides  of 
a  triangle  are  given,  the  angles  can  be  found  directly  from  the 
formulas  proved  in  Art.  96,  p.  133. 

*•  <« 


In  order  to  obtain  a  form  suitable  for  computation  by  means 
of  logarithms  we  proceed  as  follows : 

Let  the  sum  of  the  sides  of  the  triangle  #  +  £>-h<?=2s. 
Then  we  have  a  +  £_  c  =2  (s—  <?) 

b  +  e  —  a  =  2  (s  —  a), 

Then,  1  —  cos  A  =  I  — 


2  be 


2  be 


2  be 
b  —  c)(a  —  b  +  c) 

2  be 

-b}(s-e) 
be 


THE   OBLIQUE   TRIANGLE  •  149 

Also  (Art.  82,  p.  108),      1  -  cos  A  =  2  sin2  ^. 


NOTE.     Since  A  <  180°,  being  one  of  the  angles  of  a  triangle,       <  90°  ; 

\  A  A 

therefore  sin  —  ,  cos—,  and  tan—  are  positive.     Hence  the  radical  in  (4), 
and  the  corresponding  expressions  in  (5)  and  (6)  below,  are  always  positive. 

-,    ,    52  4-  c2  —  a? 
In  like  manner,  1  +  cos  A  =  1  H 


2  be 


2  be 


2  be 


2  be 


_  2  s(s  -  a) 
be 

Also  (Art.  82,  p.  108),       1  +  cos  A  =  2  cos2^. 


COS 


ca 


(7 

o  =  \ 
* 


. 
—  6) 


Dividing  (4)  by  (5),  we  have 

tap:f =  \      7(g  -a)  ^ 

In  like  manner  it  can  be  proved  that 


150  •  PLANE   TRIGONOMETRY 

Any  one  of  the  three  formulas  just  given  can  be  used  in 
finding  the  angle  required.  If  the  half  angle  is  very  small,  the 
cosine  formula  will  not  give  a  result  as  accurate  as  either  the 
sine  formula  or  the  tangent  formula,  since  the  cosines  of  angles 
that  are  very  small  differ  but  little  from  each  other ;  and  for 
a  similar  reason  the  sine  formula  should  not  be  used  when  the 
half  angle  is  near  90°.  In  general  the  tangent  formula  is  better 
than  either  of  the  others. 

To  insure  as  great  a  degree  of  accuracy  as  possible,  it  is 
better  to  solve  for  all  the  angles  rather  than  solve  for  two 
angles  and  then  subtract  their  sum  from  180°.  If  each  angle 
is  computed  separately  and  their  sum  is  found  to  be  within 
two  or  three  seconds  of  180°,  the  work  of  solution  is  probably 
correct. 

If  all  the  angles  are  to  be  computed,  the  following  variation 
of  the  tangent  formula  may  be  found  useful. 


tan      = 

2       *  s(s-a)* 


1       /( s  —  a )  ( s  —  b)(s  —  c) 

~~ 


Putting  V-          ~  =  *•• 

we  have  Uni  =  7^' 

In  like  manner,  tan  —  = — ;  (9) 

2      s  —  b 

tan  !=-£-.  (10) 

Ex.1.     Given    a  =  79. 3,    5  =  94.2,    c>=66.9;    find   all   the 
angles. 

The  work  of  solving  for  A  and  B  is  as  follows  : 

a  =79.3  s-  a  =  40.9 

b  =  94.2  s  -  b  =  26 

c  =  66.9  s-c  =  53.3 
2  s  =  240.4  s  =  120.2 

s  =  120.2 


THE   OBLIQUE   TRIANGLE 


151 


log  (*-&)=  1.41497 

log  0  -  c)  =  1 .72673 
colog  (s-  «)  =  8.38828  -10 
colog  6-  =  7.92010-10 
2)19.45008  -20 

log  tan  ^  =  9.72504-10 

.-.  |-  =  27°57'56". 
A  =  55°  55'  52". 


log  (s-c)=  1.72673 
log  (s- a)  =1.61172 
colog  (s  -  b)  =  8.58503  -  10 
colog  .s=^.92010 -10 

2)19.84358-20 
log  tan— =  9.92179-10 

...  ^  =  39°  52' 6.9". 

B  =  79°  44'  13.8". 

^+B  =  135°40'5.8'/. 

.-.  (7  =  44°  19' 54.2". 


If  the  value  of  C  is  found  by  logarithms  in  the  same  manner  as  were  the 
values  of  A  and  B,  it  will  be  found  to  be  44°  19'  56.8",  which  is  2.6"  larger 
than  the  value  found  by  subtracting  the  sum  of  A  and  B  from  180°.  The 
sum  of  the  three  angles,  when  all  are  found  independently,  is  180°  0'  2.6". 
The  sum  of  the  three  angles  determined  in  this  manner  is  rarely  equal  to 
exactly  180°.  This  is  due  to  the  fact  that  logarithmic  computation  is 
almost  always  slightly  inexact.  It  is  customary  in  practical  work  to  divide 
the  error  among  the  three  angles  according  to  the  probable  amount  for  each 
angle. 

Ex.  2.  Solve  the  preceding  example  by  the  use  of  formulas 
(8),  (9),  arid  (10). 

In  solving  by  this  method  it  is  best  to  find  all  the  logarithms 
at  the  outset,  and  then  to  subtract  the  logarithms  of  s  —  a, 
s  —  b,  s  —  c,  respectively,  from  the  logarithm  of  r.  A  com- 
pact arrangement  of  the  work  can  be  secured  by  following  the 
model  below. 

log  (s-  a)  =  1.61172 
log  (s-  b)  =1.41497 
log  (s-c)  =  1.72673 

colog  s  =  7.92010  -  10 
log  r2  =  2.67352 
log  r=  1.33676 


s  =  120.2  Check. 


Check. 


log  tan  ^  =  9.72504 -10 
log  tan  |  =  9.92179  -  10 
log  tan  =  9.61003 -10 


A 

2 
B_ 

2 
C  _ 
2 
A  = 
B  = 
C  = 

27°  57' 
39°  52' 

22°  9' 

55°  55' 
79°  44' 
44°  19' 

56" 
6.9" 

58.4" 

52" 
13.8" 
56.8" 

152  PLANE   TRIGONOMETRY 

EXERCISE  XXVII 

1.  Given  a  =  56,  ft  =  58,  c  =  64  ; 

find     ^=54°  22'  43",  £=  57°  20'  32",  (7=  68°  16'  44". 
If  fe^  '  1 

2.  Given  a  =  54,  5  =  52,  e  =  68  ; 

find    4  =  51°  24'  3.8",  B  =  48°  48'  52.8",  O=  79°  47'  7.6". 

3.  Given  a  =  35,  ft  =  41,  c  =  47  ; 

find     .4  =  46°  15'  5",  £=57°  48'  16",  C  =  75°  56'  41.5". 

4.  Given  a  =  73,  b  =  82,  c  =  91  ; 

find     A  =  49°  34'  58",  ^=58°  46'  58",  C=71°38'4". 

5.  Given  a  =  47,  ft  =  51,  c  =  58; 

find     .4  =  50°  35'  18",  .B  =  56°  58'  4",  6Y=  72°  26'  40". 

6.  Given  a  =  286,  ft  =  321,  c  =  463  ; 

find     J.  =  37°  33'  57",  £  =  43°  10'  46",  G7=  99°  15'  23". 

7.  Given  a  =  138,  ft  =  246,  c  ==  321  ; 
find     ^=23°  47'  23",  ^=45°  58'  41",  0=110°  14' 

8.  Given  a  =  196,  ft  =  211,  <?=173; 
find     vl  =  60025'31",  £=69°  26',  6^=50°  8'  36". 

9.  Given  a  =  48.3,                 ft  =  53.2,  <?  =  62.7; 
find     ^  =  48°  24'  24",  ^=55°  27'  44",  C=  76°  7'  55". 

10.    Given  a  =  226.4,  ft  =  431.  6,  c=  316.8; 

find     ^=30°  35'  53",  5=103°  58'  55"  C  =45°  25'  8". 

/ll.    Given  a  =  262.43,  ft  =  514.36,  c  =  556.25  ; 

find     A  =  50°  59'  18"  £ 


12.    Given  a  =  2243.  8,  ft  =  2469.2,  c  =  3125.6; 

find     ^1  =  45°  26'  3",      ^  =  51°  37'  42",       (7=  82°  56'  19". 


\  f ' 


' 

-  7 


THE   OBLIQUE   TRIANGLE  153 


13.  Given  a  =  25617,  6  =  34178,  c  =  23194; 
find    .4  =  48°  31'  56",    5=  88°  44'  34",       (7=  42°  43'  30". 

14.  Given  a  =  0.34177,          b  =  0.45623,  c  =  0.58216  ; 
find     A  =  35°  54'  30",    B  =  51°  31'  34",       0  =  92°  33'  56". 

15.  Given  a  =  11.682,  £  =  14.468,  c=  20.386; 
find    ^  =  34°  6'  13",       £  =  43°  58'  47",       (7=  101°  54'  58". 

16.  Given  a  =  1.9141,  6  =  1.8365,  c=  1.2854; 

find    A  =  73°  14'  32,"    B  =66°  44'  22",       <7=40°1'5". 

17.  The  sides  of  a  triangle  are    respectively  36.92,  31.84, 
26.14.     Find  the  smallest  angle  of  the  triangle. 

18.  The  sides  of   a  triangle  are  in  the  ratio  of  29  :  21  :  38. 

Find  the  medium  angle.  |>t  \\±^^  ({ 

19.  The  sides  of  a  triangle  are  to  each  other  as  3:4:5.    Find    \(LI!  -   (& 

all  the  angles. 

i  S   ^  ~~  ~~ 

20.  In  a   given  triangle  a  =8,   6  =  8,  c  =  S.     Find  all  the 

angles. 

21.  Three  cities  are  respectively  22.6,  21.4,  19.6  miles  apart. 
If  the  curvature  of  the  earth  is  disregarded,  what  angles  are 
made  by  the  lines  joining  the  cities? 

22.  In  discussing  the  solution  of  a  triangle  when  two  sides 
and  the  angle  opposite  one  of  them  are  given,  it  was  noted  that 
two  solutions  were  possible  when  an  angle  was  found  by  means 
of  its  sine.      Why  does  not  a  similar  ambiguity  exist  when  an 
angle  is  found  by  means  of  formula  (4),  p.  149? 

23.  The  sides  of  a  triangle  are  a  =  7,  b  =  8,  c  —  5.     Find  the 
angle  A. 

24.  The  sides  of  a  triangle  are  a  =  7,  6  =  5,  c  —  3.     Find  the 
angle  A. 

25.  An  object  16.2  ft.  in  length  is  so  situated  that  one  end 
is  17J  ft.  and  the  other  is  11.9  ft.  from  the  eye  of  an  observer. 
What  angle  does  the  object  subtend  at  the  eye? 

ff#  -' 

>i     I  a  *»    — 


154 


PLANE    TRIGONOMETRY 


103.  Area  of  a  triangle.      In  geometry  it  was  proved  that  the 
area  of  a  triangle  (A)  can  be  found  by  either  of  the  following 
formulas:  A  =  |  base  x  altitude, 

or,  A  =  Vs  (s  —  a)(s—  b)(s  —  <?). 

The  work  of  finding  the  area  of  a  triangle  can  be  greatly 
simplified  by  trigonometry,  as  will  be  seen  from  the  following 
section. 

104.  CASE  1.     Given  two  sides  and  the  included  angle.     The 
area  of  any  triangle  is  equal  to  one  half  the  product  of  the  base 
and   the   altitude.     Therefore,  using   either  of   the  following 
figures, 


But 


A  =  1  c  CD. 
CD  =  a  sin  B. 


c  — 


sin  A. 


Substituting  this  value  of  c  in  (1),  Case  1,  we  have. 

a2  sin  B  sin  C 


A  = 


2  sin  A 


But  since  A  +  B  +  6'=  180°,  sin  A  =  sin  (B  +  <7)  ; 

.  ^  __   a2  sin  B  sin  C 
=  2  sin  (B  +  (7) 


CD 

(2) 


In  like  manner  it  can  be  proved  that 
A  =  *  be  sin  A, 
and  A  =  -*-  ab  sin  C. 

CASE  2.     Given  a  side  and  the  two  adjacent  angles.     By  the 
law  of  sines  (Art.  94,  p.  131), 

a  :  c  =  sin  A  :  sin  C. 
a  sin  C 


(4) 


THE   OBLIQUE   TRIANGLE  155 

CASE  3.     Given  the  three  sides.     In  Art.  80,  p.  106,  it  was 
proved  that  ,          , 

sin  A  =  2  sin  —  cos  — . 

L  '— 

But  (Art.  102,  p.  149), 


be 
and  cos  -^  ' 

Substituting  these  values  in  the  above  equation,  we  have 


2 
sin  A  =  T-S(S  —  a)(s—b)(s  —  c). 


Substituting  this  value  of  sin  A  in  (2),  we  have 


A  =  V*(«  -«)(«_  6)(s  -  c).          (5) 

CASE  4.     Given  three  sides  and  the   radius  of  the  circum- 
scribed circle.     By  Art.  95,  p.  132,  we  have 


where  r  is  the  radius  of  the  circumscribed  circle.     Substituting 
this  value  in  (2),  we  have 

A=f£  (6) 

CASE  5.  Given  three  sides  and  the  radius  of  the  inscribed 
circle. 

Let  r  be  the  radius  of  the  inscribed  circle.  The  triangle 
can  be  divided  into  three  triangles  whose  bases  are  a,  6,  <?,  re- 
spectively, and  whose  common  altitude  is  r.  Then 

A  =  £r(«  +  ft  +  c).  (7) 


CHAPTER   XIII 

MISCELLANEOUS   PROBLEMS   IN   HEIGHTS  AND 
DISTANCES 

105.  In  this   chapter   certain   problems   will  be  considered 
that   are   frequently  met    in    land   surveying,   railroad   work, 
etc.     The  degree  of  accuracy  required  in  practical  problems  of 
this  kind  can  only  be  known  after  the  nature  of  the  special 
problem  under  consideration  is  known.     Hence,  in  the  examples 
that  are  here  considered   no  attempt  is  made  to  conform  to 
the  ordinary  practice  of  field  surveyors.     In  many  classes  of 
problems  that  they  are  called  upon  to  solve  a  .sufficient  degree 
of  accuracy  is,  secured  if  the  angles  are  measured  to  single 
minutes  and  the  computations  are  performed  by  means  of  four- 
place  tables  of  logarithms;  while  in  others  the  measurements 
are  made  with  the  greatest  possible  accuracy  and  the  computa- 
tions are  performed  with  the  aid  of  eight-,  ten-,  or  twelve-place 
tables.     For  this  reason  it  is  quite  impracticable  for  an  elemen- 
tary text-book  in  trigonometry  to  attempt  to  conform  to  field 
usage. 

The  tables    used  in  the  solution  of    the    problems  in   this 
chapter  are  five-place  tables. 

106.  The  height  of  an  object  by  means  of  observations  made 
at  distant  points. 

Let  AB  represent  the  height  of  an  object,  and  let  (7,  JJ, 
be   two  points  of  observation   on  the   same   level  with  A,  so 

situated  that  A,  0,  Z),  are  in  the 
same  straight  line.  Let  the  angle 
of  elevation  of  B  at  C  be  «,  and  at 
D  be  /3,  and  let  DC=a.  Then  from 
the  triangle  ABC 

./  •  /  ~t  *\ 

/,v<  =  s1""' 

150 


PROBLEMS   IN    HEIGHT-  L  FIANCES  157 


and  from  the  triangle  DCB 
BC 


a        sin  (a— 


Substituting  this  value  of  1M?  in  (1).  and  reducing  we  have 


a  formula  which  gives  the  value  of  x  in  a  form  suitable  for 
computation  either  by  logarithms  or  by  natural  functions. 

Ex.  1.  What  is  the  height  of  a  tower  when  the  angles 
of  elevation  of  the  top  of  the  tower  from  two  points  250  ft. 
apart  on  the  ground  and  in  the  same  straight  line  with  the  foot 
of  the  tower  are  30°  and  60°  respectively? 

=  60%  and  0  =  3G°.    Therefore 
>in  60°  sin  30° 


x  = 


sin  3IT 
50-1  x^  =  218.5  ft. 

107.  If  the  height  of  an  object  is  to  be  determined,  and  no 
two  points  can  be  found  that  are  in  the  same  st  might  line, 
and  at  ilie  same  time  conveniently  situated  for  observation,  the 
following  method  is  often  employed  : 

From  A  measure  AB  in  any  convenient  direction.  Let  the 
angle  of  elevation  of  the  top  of  the  object  D,  measured  at  A* 
be  «,  and  let  the  distance  AB  be  a.  At  A  and  B  measure  the 
angles  DAB=&  and  DIM  =  7,  respectively.  Then  in  the 
triangle  AD* 


Therefore, 

AD  _  _  81117  _  _       siu  7 
~o~  "  sin  (1W  -  (£  +  7))  ~  sin  (£  +  7) 

1  -ing  the  value  of  AD  obtained 
from  this  equation,  we  have 


. 
sin  (£  +  7) 


158  PLANE  TRIGONOMETRY 

MISCELLANEOUS   EXAMPLES 

THE    RIGHT    TK I  ANGLE 

1.  The   angle   of   elevation    of    the    top  of   a   vertical   cliff 
426.28  ft.  high,  taken  from  a  point  on  the  same  level  as  the 
foot  of  the  cliff,  is  59°  51'  14".      What  is  the  distance  from  the 
foot  of  the  cliff  to  the  point  where  the  observation  was  taken? 

2.  A  pole  36  ft.  high  casts  a  shadow  39  ft.  long.      What  is 
the  angle  of  elevation  of  the  top  of  the  pole,  measured  at  the 
extremity  of  the  shadow? 

3.  The    height   of   a   room   is    12.62   ft.    and    its   length  is 
14.44  ft.     What  is  the  angle  of  e!«  of  one  of  the  upper 
corners  of  the  room  taken   at  the  lower   corner   on  the  same 
side? 

4.  What  is  the  elevation  of  the  sun  when  a  tree  31.6  ft.  high 
casts  a  shadow  42.9  ft.  in  length  ? 

5.  What  angle  does  a  ladder  25.2  ft.  long  make  with  the 
ground  when  it  just   reaches  the  sill  of   a  window  18.95  ft. 
above  the  level  on  which  the  foot  of  the  ladder  rests  ? 

6.  The  angle  of  depression  of  a  point  on  the  ground,  meas- 
ured from  the  top  of  a  building  49.27  ft,  high,  is  34°  6'  36". 
What  is  the  distance  from  the  foot  of  the  building  to  the  given 
point  ? 

7.  The   length   of   the   diagonal    of    a   rectangular   field   is 
247.39    ft.,    and    the    angle    between    the    diagonal    and    the 
shorter  side  of  the  field  is  29°  40'    36".      What  is  the  width 
of  the  field? 

8.  A  path  measuring  256.4  ft.  in  length  leads  diagonally 
across  a  rectangular  plot  of  ground,  making  with  one  of  the 
sides  an  angle   of   61°  12'  52".     What  is  the    length   of   the 
side  ? 

9.  The  angle  of  elevation  of  a  balloon  measured  at  a  certain 
point  is  71°  14'  12",  and  from  this  point  to   a  point  directly 
below  the  balloon  the  horizontal  distance  is  415.9  ft.     What 
is  the  height  of  the  balloon  and  its  distance  from  the  point  of 
observation  ? 


PROBLEMS    IX    HEIGHTS   AND   DISTANCES  159 

10.  A  kite  is  fastened  to  a  string  483.2  ft.  long,  and  the  string 
makes  an  angle  of  63°  19'  28"  with  the  level  of  the  ground. 
What  is  the  vertical  height  of  the  kite  above  the  ground,  no 
allowance  being  made  for  the  sagging  of  the  string  ? 

11.  To  ascertain  the  width  of  a  river  a  distance  AB  is  meas- 
ured along  one  of  the  banks  262.38  ft.     Directly  across  the 
river  from  B  is  a  point  (7,  and  the  angle  BAG  is  found  upon 
measurement  to  be  41°  38'  20".     Required  the  width  of  the 
river. 

12.  Two  forces,  of  198.52  Ib.  and  393.13  Ib.  respectively, 
are  acting  at  right  angles  to  each  other.    What  is  the  resultant 
of  the  two  forces,  and  what  is  the  angle  which  the  direction 
of  each  force  makes  with  the  resultant  ? 

13.  What  is  the  radius  of  the  parallel  passing   through  a 
point  on  the  earth's  surface  whose  latitude  is  43°  15',  the  radius 
of  the  earth  being  reckoned  as  3956  mi.  ? 

14.  The  angle  of  elevation  of  the  top  of  aitill  viewed  from 
a  certain  point  is  29    17',  and  from  a  point  362.4  ft.  nearer, 
measured  directly  toward   the  hill,  the  angle  of  elevation  is 
48°  12'.     Required  the  height  of  the-MH-.  -•£ 

15.  From  the  top  of  a  mountain  the  angles  of  depression  of 
two  milestones  2  mi.  apart  and  in  the  same  vertical  plane  with 
the  top  of  the  mountain  are  10°  14'  42"  and  5°  38'  46"  respec- 
tively.    What  is  the  height  of  the  mountain? 

16.  A  flagstaff  which  is  broken  at  a  certain  distance  above 
the  ground  falls  so  that  its  tip  touches  the  ground  at  a  distance 
of  13.5  ft.  from  the  foot  of  the  portion  which  remains  standing. 
The  length  of  the  part  broken  over  is  35.1  ft.     What  was  the 
total  height  of  the  staff  before  it  was  broken  over  ? 

17.  If  the  angle  of  depression  of  the  visible  horizon,  observed 
from  the  top  of  a  mountain  3  mi.  in  height,  is  2°  13'  59",  what  is 
the  diameter  of  the  earth  ? 

18  A  ladder  30  ft.  long  when  set  in  a  certain  position 
between  two  buildings  will  reach  a  point  20  ft.  from  the 
ground  on  one  of  the  buildings,  and  on  being  turned  over 
without  having  the  position  of  its  foot  changed  it  reaches  a 


1HO  PLANE   TRIGONOMETRY 

point  on  the  other  building  15  ft.  from  the  ground.  What  is 
the  angle  between  the  two  positions  of  the  ladder  ?  (Solve  by 
natural  functions.) 

19.  A  lighthouse  50  ft.  high  stands  on  the  top  of  a  rock. 
The  angle  of  elevation  of  the  top  of  the  rock  and  of  the  top 
of  the  lighthouse,  measured  from  the  deck  of  a  vessel,  are  6°  5' 
and  6°  58"  respectively.     What  is  the  height  of  the  rock,  and 
the  distance  from  the  vessel  to  the  foot  of  the  rock  ?     (Solve 
by  natural  functions.) 

20.  At  any  point  on  the  earth's  surface  a  line  is  drawn  tan- 
gent to  the  surface  at  that  point.     If  the  earth  is  considered  a 
sphere  whose  diameter  is  7912.4  mi.,  how  far  from  the  surface 
will  the  line  be  at  the  end  of  1  mi.? 

21.  A  building  50  ft.  high  stands  at  the  foot  of  a  hill,  and 
from  the  top  of  the  hill  the  angles    of  depression  of  the  top 
and  of   the   bottom   of   the  building  are  45°  15'  and  47°  12' 
respectively.     What  is  the  height  of  the  hill  ? 

22.  The  angles  of  a  triangle  are  1:2:3,  and  the  perpendicu- 
lar  from    the   greatest  angle    to    the    side  opposite   is  15   ft. 
Required  the  sides  of  the  triangle. 

23.  A  bridge  of  five  equal  spans  crosses  a  river,  each  span 
measuring  100  ft.  from  center  to  center.      From  a  boat  moored 
in  line  with  one  of  the  middle  piers  the  length  of  the  bridge 
subtends  a  right  angle.     What  is  the  distance  from  the  boat  to 
the  bridge?     (Solve  by  natural  functions.) 

24.  An  observer  on  a  vessel  at  anchor  sees  another  vessel 
due  north  of  him;    at  the  end  of  fifteen  minutes  it  bears  E., 
and  half  an  hour  later  it  bears  S.E.      How  long  after  it  is  first 
seen  is  it  nearest  the  observer,  and  in  what  direction  is  it  sail- 
ing, its  course  being  supposed  to  be  in  a  straight  line  from  the 
time  of  the  first  to  the  time  of  the  last  observation?    (Solve  by 
natural  functions.) 

25.  A  statue  on  a  column  subtends  the  same  angle  at  dis- 
tances of  27  and  of  33  ft.  from  the  column.     If  the  tangent  of 
the  angle  equals  T^,  what  is  the  height  of  the  statue  ?     (Solve 
by  natural  functions.) 


PROBLEMS   IN   HEIGHTS    AND   DISTANCES  161 

26.  A    tower    145   ft.  high  stands  on  an  elevation    75    ft. 
high.     At   what    point  in   the   plain   on   which  the   elevation 
stands  must  an  observation  be  made  in  order  that  the  tower 
and  the   height  of  the   elevation   may  subtend  equal  angles? 
(Solve  by  natural  functions.) 

27.  A  flagstaff  50  ft.   high  stands  in  the  center  of  a  plot 
of  ground  in  the  form  of  an  equilateral  triangle.      Each  side 
of  the  triangle  subtends  at  the  top  of  the  staff  an  angle  of  60°. 
What  is  the  length  of  one  of  the  sides  of  the  triangle  ?     (Solve 

by  natural  functions.)  * 

28.  A    tower    stands    on   the    slope  of  a   hill   that   has   an 
inclination  of  15°  to  the  level  of  the  plain.      At  a  point  80  ft. 
farther  up  the  hill  it  is  found  that  the  tower  subtends  an  angle 
of  30°.    Prove  that  the  tower  is  40(V«J-  V£)  ft.  in  height. 

29.  At  a  distance  of  300   ft.  from  the  foot  of  a  tower  the 
angle  of  elevation  is  one  third  as  great  as  it  is  at  a  distance  of 
60  ft.     What  is  the  height  of  the  tower? 

THE    OBLIQUE    TRIANGLE 

30.  The  angles  of  elevation  of  a  balloon  measured  at  the 
same  instant  at  two  points  on  level  ground  and  in  the  same 
vertical  plane  as  the  balloon  are  41°  56'  and  28°  14'  respectivel}T. 
The  two  points  from  which  the  angles  are  measured  are  3462 
ft.  apart  and  on  the  same  side  of  the  balloon.      Required  its 
height  at  the  time  of  observation. 

31.  The  angle  of  depression  of  an  object  viewed  from  the 
top  of  a  tower  is  50°  12'  56",  and  the  angle  of  depression  of 
a  second  object  250  ft.  farther  away,  and  in  a  straight  line  with 
the  first  object  and  the  foot  of  the  tower  is  31°  19'  54".     What 
is  the  height  of  the  tower  ? 

32.  The  angles  of  depression  of  two  objects  on  a  level  plain, 
viewed  from  an  elevation  in  the  same  vertical  plane  with  the 
objects,  are  48°  12'  and  29°  17'  respectively,  and  the  distance 
between  the  two  points  is  362.4  ft.     Required  the  height  of 
the  point  of  observation. 

CON  ANT'S  TKIG.  —  11 


162  PLANE   TKIGONOMETKY 

33.  The  sides  of  a  triangular  plot  of  ground  are  138  ft., 
246  ft.,  and  321  ft.  respectively.     What  is  the  greatest  angle 
formed  by  the  sides? 

34.  Two  objects  are  separated  by  a  building,  and  it  is  re- 
quired to  find  the  distance  between  them.     At  a  third  point, 
distant  268  ft.    and  315  ft.  respectively  from  the  given  ob- 
jects, the  angle  subtended  by  the  line  connecting  the  objects 
is  measured  and  is  found  to  be  108°  17'.    What  is  the  distance 

between  the  objects  ? 
•i^. 

35.  What  is  the  distance  between  two  points  4,  B,  when 

the  distances  from  these  points  to  a  third  point  C  are  6282  ft. 
0       and  2344  ft.  respectively,  and  the  angle  :S*UL  is  119°  40'  40"? 
Is  more  than  one  solution  possible  ?     Why  ?     (See  Art.  100, 
p.  138.) 

36.  The  distance  between  two  points  A,  B,  cannot  be  ob- 
tained directly  by  the  use  of  the  chain  or  tape  because  of  an 
intervening  body  of  water.     A  third  point   C  is  chosen  from 
which  both  A  and  B  are  visible,  and  the  following  measure- 
ments   are    then  made:      4(7=3101.8  ft.,    Z  CAB  =  51°   28', 
Z.  ABQ  =  70°  37'  33".     What  is  the  required  distance  ? 

37.  In  a  system  of  triangulation  the  sides  of  a  triangle  con- 
necting the  stations  on  the  tops  of  three  hills  have  been  com- 

c  puted  and  have  been  found  to  be  54,692.73  ft.,  61,284.39  ft., 
and  42,798.64  ft.  respectively.  What  are  the  values  of  the 
angles  of  this  triangle  as  computed  from  the  sides  ? 

38.  An  observation  station  A  is  set  up  in  a  field  along  one 
side  of  which  runs  a  straight,  level  road.     Two  points  of  ob- 
servation on  the  road,  J5,   (7,  one  fourth  of  a  mile  apart,  are 

*o  chosen,  on  opposite  sides  of  the  first  station  and  the  angles 
ABO,  ACB,  are  measured  and  found  to  be  46°  20'  28"  and 
38°  24'  48"  respectively.  What  is  the  distance  from  the  station 
A  to  the  road  ? 

39.  The  distances  from  a  point  on  shore  to  two  buoys   are 
known  to  be  1286  ft.  and  2466  ft.  respectively,  and  the  angle 
subtended  at  that  point  by  the  line  connecting  the  buoys  is  42° 
14'  16".      What  is  the  distance  between  the  buoys? 


PROBLEMS  IN   HEIGHTS   AND   DISTANCES  163 

40.  A  tripod  is  set  up  on  a  rock,  and  to  find   the   distance 
from  the  tripod  to  the  shore  a  line  8500  ft.  in  length  is  meas- 
ured along  the  shore,  and  at  each  extremity  of  the  line   the 
angle   is   measured    which   subtends   the    line    connecting  the 
tripod  with  the  other  end  of  the  line.     The  angles  are  found 
to  be  46°  28'  and  43°  32'  respectively.    Find  the  distance  from 
the  tripod  to  the  line  of  measurement  along  the  shore. 

41.  Two  vessels  lying  at  anchor  1  mi.  apart  are  observed 
from  a  third  vessel  sailing  east  to  be  in   a  straight  line  due 
north.     After  sailing  an  hour  and  a  half  one  of  the  vessels 
bears  N.W.  and  the  other  W.N.W.     Find  the  rate  at  which 
the  vessel  is  sailing.  \  «    \  ^  ^ 

42.  The  distance  between  two  points  A,  B,  is  to  be  deter- 
mined, where  B  is  accessible  and  -4Ms  not.     A  kite  is  sent  up 
and  made  fast,  and  its  position   0  is  determined  to  be  517.3 
yd.  vertically  above  D,  which  is  on  the  same  level  with  A  and 
B.     The  following  angles  are  then  measured:     A 6^5  =  13°  15' 
15",   CAD=  21°  9'  18",  DBC=<2&°  15'  34". \What  is  the  dis- 
tance from  A  to  B?  J    $v  %    $~  I 

43.  Two  forces,  of  410  Ib.  and  320  Ib.  respectively,  are  act- 
ing at  an  angle  of  51°  37'.     Required  the   direction  and  in- 
tensity of  the  resultant. 

44.  A  kite  A  has  been  sent  up  and  is  fastened  to  the  ground 
at  a  point  Q.     The  kite  has  drifted  a  certain  distance  and  now 
stands  directly  above  a  point  B,  which  is  on  the  same  level  as 
(7,  but  is  separated  from  it  by  obstacles  which  render  direct 
measurement  impracticable;  and  the  height  of  the  kite  is  de- 
sired.    To  ascertain  this  a  line  is  measured  from  0  to  a  point 
Z),  4262.4  ft.    in  length,  and    the    following   angles  are  meas- 
ured:   ACB=  31°  17'  14",    ACD=  66°  14'  52",  CDA  =  52°  51' 
38".   Required  the  vertical  height  of  the  kite  above  the  point  B. 
(See  Art.  107.) 

45.  Two  rocks  are  to  be  charted.     To  ascertain  the  distance 
between  them  the  angles  of  elevation  of  a  point  at  the  top  of  a 
cliff  527.4  ft.   high   are   taken  and  are  found  to  be  21°  8'  16" 
and  23°  14'  20"  respectively,  and  the  angle  subtended  by  the 


164  I'LAM-:    TUGOXOMHTKY 

line  connecting  the  rocks,  measured  at  a  point  at  the  top  of 
the  cliff,  is  16°  3'  30".  Required  the  distance  between  the 
rocks. 

46.  A  balloon,  J.,  is  sighted   at  the  same  instant  from  two 
points,  B,  C,  which  are  on  the  same  level,  and  are  262.4  ft. 
apart.     The  angle  of  elevation  of  the  balloon  at  B  is  41°  15' 
24",  ZABC  =  62°  48'  14",  ZACB=59°  14'  21".    What  is  the 
height  of  the  balloon  at  the  instant  of  observation?     ^  ^\  ^  t  *^  \ 

47.  A  tower  stands  on  the  slope  of  a  hill  which  makes  an 
angle  of  16°  with  the  horizon.      At  a  distance  of  95  ft.  from 
the  foot  of  the  tower,  measured  directly  up  the  side  of  the  hill, 
the  height  of  the  tower  subtends  an  angle  of  38°.     What  is  the 
height  of  the  tower? 

48.  A  tree  stands  E.S.E.  of  an  observer,  and  at  noon  the 
extremit}^  of  the  shadow  of  the  tree  is  directly  N.E.    of  the 
position  in  which  he  is  standing.     The  length  of  the  shadow  is 
60  ft.,  and  the  angle  of  elevation  of  the  top  of  the  tree  viewed 
from  the  position  of  the  observer  is  45°.     What  is  the  height  of 
the  tree?     (Solve  by  natural  functions.) 

49.  It  is  required  to  find  the  distance  between  two  points, 

A,  B,  neither  of  which  is  accessible.     For  that  purpose  a  base 
line,  (7Z>,  4968  ft.  long,  is  measured,  and  the  following  angles 
are  observed:    ACD=W8°  14',   £6^Z)  =  41°  15',  £7X7=11,5° 
21',  ADO=  39°  42'.      What  is  the  distance  from  A  to  B> 

50.  Two    points    are  so  situated  that   it   is   not    possible  to 
measure  directly  from  one  to  the  other,  but  observations  can 
be  taken  at  either  point.      Two  other  points,  (7,  D,  are  chosen, 
5226  ft.  apart,  and  the  following  angles  are  measured:  ACB 
=  15°  18'  24",  DAO=  21°  12'  46",  DBC=  23°  18'  42",  AT)C  = 
BDC=  90°.      What  is  the  distance  from  A  to  #? 

51.  To  find  the  distance  between  two  inaccessible  points,  A, 

B,  two  other  points,   (7,  D,  are  chosen,'  so  situated  that  from 
either  of  them  the  three  other  points  can  be  seen;  and  the  fol- 
lowing measurements  are  then  made:    6Y.Z)  =  826.5  ft.,  ZACD 
=  121°  12',  Z BOD  =  58°  55',  Z.ADC=  49°  12',  Z.ADB  =  §2° 
38'.      What  is  the  distance  from  A  to  B'> 


PROBLEMS   IN   HEIGHTS   AND   DISTANCES 


165 


52.  Two  points,  A,  B,  are  so  situated  that  only  one  point,  (7, 
can  be  found  which  is  conveniently  situated  for  observation, 
from  which  both  can  be  seen.  A  fourth  point,  D,  is  found  from 
which  A  and  Q  can  be  seen,  and  a  fifth  point,  J£,  from  which  B 
and  C  can  be  seen.  The  following  measurements  are  taken, 
from  which  it  is  required  that  the  distance  from  A  to  B  shall 
be  computed:  CD  =  6428.72  ft.,  OE  =  5872.54  ft., 


=  64°  21'. 

53.  Two  points,  A,  B,  are  so  situated  that  no  point  can  be 
found  from  which  both  can  be  seen.  Two  other  points,  (7,  1),  are 
found,  so  placed  that  A  and  D  can  be  seen  from  C  and  B  from  D, 
and  also  two  additional  points,  E,  F,  so  placed  that  A  and  0 
can  be  seen  from  F,  and  B  and  D  from  E.  The  following  data 
can  now  be  obtained  for  the  determination  of  the  distance  from 
A  to  B:  (7Z)=1254  ft,,  OF  =1216  ft.,  7)^=1216  ft.,  Z.AFC 
=  78°  14'  15",  ZFCA  =  53°  51'  40",  Z.4CZ)  =  52°  17'  18",  Z.  CDB 
=  155°  24'  20",  ZBDE=53°  49'  8",  ZD^  =  82°  57'.  What 
is  the  length  of  the  line  AB? 


CHAPTER   XIY 


FUNCTIONS    OF    VERY    SMALL     ANGLES  —  HYPERBOLIC 
FUNCTIONS  —  TRIGONOMETRIC   ELIMINATION 

108.  Trigonometric    functions    of    very    small    angles.     Let 
A  OB  be  any  angle  less  than  90°*     With  0  as  a  center  and  any 

radius  OA  describe  a  circle. 

Draw  BQ  perpendicular  to  OA,  and 
produce  it  to  intersect  the  circle  in  B1 ' . 

Draw  tangents  to  the  circle  at  B,  B'. 
These  tangents  will,  by  geometry,  inter- 
sect OA  produced  in  the  same  point  D. 
Then 

chord  BB'  <  arc  BB'  <  BD  +  B' D. 
Dividing  by  2,        CB  <  arc  AB  <  DB. 

CB      arc  AB      BD 
'OB          OB      "  OB ' 

CB       .    /j     BD  ;\rcAB  , 

But    -— —  =  sm0,     — —  =  tan  0,    and  =  the    circular 

OB  OB  OB 

measure  of  the  angle  0,  or  of  the  arc  AB  (Art.  13,  p.  16). 
Therefore,  sin  0  <  0  <  tan  0. 

This  important  result  may  be  expressed  as  follows  : 
When  6  <  90°,  sin  6,  6,  and  tan  0  are  in  the  ascending  order 
of  magnitude. 

109.  Dividing  the  inequality  just  obtained  by  sin  0,  we  have 

0 


1< 


sin<9 


<  sec  0, 


or, 


166 


FUNCTIONS  OF  VERY  SMALL  ANGLES       167 

Therefore,  —  —  lies  between  1  and  cos  6  for  all  values  of  6 
u 


between  0  and  -  • 

But  as  0  approaches  0  as  its  limit,  cos  0  approaches  1  as  its 
limit;  and  at  the  same  time approaches  1  as  its  limit. 

COS0 

Therefore,  when  0  is  very  small,  and  is  approaching  0  as  its 

limit,  — —  lies  between  1  and  a  quantity  that  may  be  made 
0 

to  differ  from  1  by  a  quantity  e  which  may  be  made  as  small  as 
we  please ;  and  as  0  approaches  0  as  its  limit,  e  also  approaches 
0  as  its  limit.  t,  .  *  ; ,  t-  ^<(  l^  e^  .  &-^ 

In  other  words,  when  flx  approaches  0  as  its  limit,  S1"     ap- 

u 

proaches  1  as  its  limit.  This  fact  is  often  expressed  by  the 
statement  that  when  0  is  very  small,  sin  0  =  0,  approximately. 

In  like  manner  it  can  be  shown  that  as  0  approaches  0  as  its 
limit,  tan  0  will  also  approach  the  limit  0 ;  that  is,  when  0  is 
very  small,  tan  0  =  0  approximately. 

From  the  above  it  follows  also  that  when  0  is  very  small,  sin 
0  =  tan  0,  approximately. 

In  this  discussion  it  should  be  remembered  that  0  is  ex- 
pressed in^circular  measure ;  i.e.  0  is  the  number  of  radians 
in  the  angle  or  arc  under  consideration. 

EXERCISE  XXVHI 

1.    Find  the  sine  and  the  cosine  of  V. 
Let  x  be  the  circular  measure  of  1'. 


Therefore,  since  x  >  sin  x  >  0,     (Art.  108) 

sin  1'  lies  between  0  and  0.000290889. 


Also,  cosl'  =  Vl-sin2l 


>  Vl  -  (0.00029088^)  2 

^O  QQQQQQQ 


>0.9999999. 

.-.cosT  =  0.9999999+.  (1) 

But  (Art.  108,  p.  166),      sin  x > x  cos  x. 

.-.  sin  l'> 0.000290888  x  0.9999999 

.kO.000290887.  (2) 


168  PLANE   TRIGONOMETRY 

Therefore,  sinl'  lies  between  (1)  and  (2);  i.e. 

sin  1'  =  0.00029088+, 
and  the  next  decimal  place  is  either  7  or  8. 

Find  approximately  the  values  of  the  following  : 

2.    sin  10'.  4.    sin  1'.  6.    cos  15'. 

S.^cosTO'.  .5.    sin  15'.  7.    sin  8". 

HYPERBOLIC   FUNCTIONS 

110.    In  the  differential  calculus  it  is  proved  that  the  follow- 
ing equations  are  true  for  all  values  of  x : 

ein  ~  _  ~        X       •    X  X>      i     .    .  .  -fl\ 

bill  X  —  JU  —  •—  -f  —         —  -T-   '       ,  {1J 

cos  ^_^_^__|_?: ^__^_...j  (2) 


where  e  =  2.7182818  •••  is  the  base  of  the  natural  system  of  log- 
arithms. In  (1)  and  (2)  x  is  the  value  of  the  angle  or  arc 
expressed  in  radians. 

If  in  (3)  x  is  replaced  by  ix,  where  i  =  V  —  1,  we  have 


X2 


The  series  in  the  first  parenthesis  is  the  same  as  the  right 
member  of  (2),  and  that  in  the  second  parenthesis  is  the  same 
as  the  right  member  of  (1).  Hence,  replacing  these  series  by 
their  values,  we  have  equation  (4)  in  the  following  form  : 

e**  =  cos  x  +  i  sin  x.  (5) 

In  a  precisely  similar  manner  it  may  be  shown  that 

e~lr  =  cos  x  —  i  sin  x.  (6) 


HYPERBOLIC    FUNCTIONS 
Adding  (5)  and  (6),  and  dividing  by  2,  we  have 

.  (7) 


Subtracting  (6)  from  (5)  and  dividing  by  2  i,  and  the  cor- 
responding value  for  sin  x  is  obtained  : 


(8) 


These  equations  give  the  values  of  the  sine  and  the  cosine  of 
any  angle  whatever  in  exponential  form. 

111.    If  in  (5)  and  (6)  of  the  preceding  section  we  replace  x 
by  ix,  the  following  equations  are  obtained  : 

e~x  =  cos  ix-\-  i  sin  ix\  (1) 

ex  =  cos  ix  —  i  sin  ix.  (2  ) 

By  addition  and  subtraction  we  obtain  from  these  the  results 

below  :  ex  ,  e-x 

cos  ix  =  -  -~  -  ;  (3) 


It  will  be  noticed  that  the  exponential  functions  which  occur 
in  the  right-hand  members  of  (3)  and  (4)  possess  a  striking 
similarity  to  those  which  appear  in  (7)  and  (8)  of  the  preced- 
ing section.  It  has  been  found  convenient  to  make  use  of  this 
similarity,  and,  corresponding  to  the  exponential  values  of 
sin  x  and  cos  x  given  in  those  equations,  to  give  the  following 
definitions : 

- — •  is  called  the  hyperbolic  cosine  of  x, 

oX  p— X 

and  is  called  the  hyperbolic  sine  of  x. 

ft 

These  functions  are  written  in  abbreviated  form  cosh  x  and 
and  sinh  x  respectively.  Accordingly  we  have 

g.r      I      g  —  X 

cosh  x——     y "      =  cos  ix ;  (5) 

sinh  x  =  —       '- —  =  — i  si  nia?.  (6) 


170  PLANE   TRIGONOMETRY 

The  name  hyperbolic  is  applied  to  these  functions  because 
they  bear  to  the  equilateral  hyperbola  a  relation  analogous  to 
that  which  sin  a;  and  cos  x  bear  to  the  circle.  (Art.  46,  p.  64.) 

The  other  hyperbolic  functions  are  denned  as  follows  : 


- 

cosh# 


sinh  x 
sech  x  =  —  1—  ;  (9) 


—  — 

cosh  x 

—  -  —  .. 
sinli  x 

112.    Ex.  l.     Prove  the  relation  sinh  0  =  0. 
By  (6),  Art.  Ill,  we  have 


(10) 


. 
2          2 

Ex.  2.     Prove  the  relation 

sinh  (x  -f  y)  =  sinh  x  cosh  y  +  cosh  x  sinh  y. 
By  definition 

sinh  (x  +  y)  =  -  i  (sin  (ix  +  iy)  ) 

=  —  i  (sin  t.r  cos  /#  +  cos  ix  sin  z/y) 

=  —  i  (i  sinh  x  cosh  ?/  +  i  cosh  x  sinh  #) 

=  sinh  x  cosh  y  +  cosh  x  sinh  ?/. 

Ex.  3.     Prove  the  relation 
sinh  x  +  sinh  y  =  2  sinh  ^^  cosh 

By  definition 

sinh  x  +  sinh  y  —  —  i  (sin  ix  +  sin  z 


HYPERBOLIC   FUNCTIONS  171 

EXERCISE  XXIX 
Prove  the  following  identities  : 

1.  cosh  0  =  1.  9.   sin  (—  ix)  =  —  sin  ix. 

.   i  TTI      .  10.    cos(  —  iz}  =  uosix. 

2.  smn  —  =  i. 

11.   tan  ix  =  i  tanh  x. 

3.  cosh—  =  0.  12-   sinh  (-2:)=  -sinh  a;. 

13.  cosh  (  —  x)  =  cosh  x. 

4.  sinh7n'=0.  ,  ,   . 

14.  coth  (  —  x)  =  —  coth  #. 

5.  cosh9r*  =  -l.  15    sech(-:r)=sech*. 

6.  sinh2mr  =  0.  16>   Csch  (  -  ^)  =  -  csch  a;. 

7.  cosh2mr  =  l.  17. 


8.  tanh  0  =  0.  18.   sech2  x  +  tanh2  x  =  1. 

19.  csch2  x  —  coth2  a;  =  —  1  . 

20.  cosh  (x  +  «/)  =  cosh  x  cosh  y  -}-  sinh  x  sinh  ^. 

21.  sinh  2x  —  2  sinh  a:  cosh  x. 

22.  cosh  2  #  =  cosh2  x  +  sinh2  #. 


23.    sinh  #—  sinh  y  =  2  cosh—  ^—^  sinh  —  H 


24.  cosh  a;  +  cosh  y=  2  cosh—  ^-^  cosh—  ~ 

2i  2 

25.  cosh  x  —  cosh  y  =  2  sinh  x     ^  sinh  'y~^  • 

2  2 

113.   The  notation  for  inverse  hyperbolic  functions  is  the 
same  as  for  inverse  circular  functions  (Art.  84,  p.  114). 

If  y  =  sinh  x, 

then,  x  =  sinli"1?/. 

But  by  (6),  p.  169,      y  =  ell. 


Solving  this  equation  for  #,  we  have 


1). 


172  PLANE   TRIGONOMETRY 


In  like  manner,  cosh'1^  =  log  (?/  -f  V«/2  —  1)  ;  (2) 

tanh-1*,  =  1  log  i±^;  (3) 

J-  —  y 

coth-1  y  =  tanh-1  -  =  \  log  £±1 ;  (4) 

y    *      y  — -* 


sechr1 «/  =  cosh  a  -  =  log *-  ;         (5) 


csch-1  y  =  sinh-1     =  logA^.         (6) 


EXERCISE   XXX 

Prove  the  following  relations  : 

1.    tanh-1  -^_  =  2  tanh-1  2:. 


2.    sinh"1  2  #  =  2  sinh"1  a;  cosh"1  a;. 


3.    sinh"1  z  =  cosh"1  Vl  +  xz. 


4.    sinh"1  a;  =  tanh"1 


VI 

T  -4- 

5.    tanh"1  x+  tann"1  y  =  tanh"1  - 


ELIMINATION 

114.  It  often  happens  that  two  or  more  equations  are  given 
that  contain  trigonometric  functions  of  some  angle,  or  perhaps 
of  more  than  one  angle.  From  these  equations  a  single  equa- 
tion is  to  be  obtained  from  which  all  trigonometric  functions 
have  been  eliminated. 

In  theory  the  required  elimination  can  always  be  performed, 
but  in  practice  this  often  involves  processes  that  are  some- 
what complicated ;  and  the  desired  results  are  obtained  with  a 
greater  or  less  degree  of  difficulty. 

No  general  rule  for  work  of  this  kind  can  be  given ;  and  the 
process  is  best  illustrated  by  a  few  examples. 


TRIGONOMETRIC    ELIMINATION  173 

115.     Ex.  i.    Find  the  values  of  r  and  6  from  the  equations 

r  sin  6  =  a  ;  (1) 

r  cos  d  =  b.  (2) 

Squaring  and  adding, 

r2  (sin2  0  +  cos2  0)  =  a2  +  6a, 
r2  =  «2  +  b2, 

r  =  \/a2  +  b*. 
Also,  dividing  (1)  by  (2), 

tan0  =  £, 

6 

0  =  tan-1  ?  . 
o 

Ex.  2.     Find  the  equation  of  relation  between  a  and  b  if 

sin3  0  =  a,     and    cos3  0  =  6. 
From  the  values  here  given  we  have 

sin  0  =  «¥,     and     cos  0  =  6*. 
But  for  all  values  of  0,       sin2  0  +  cos2  0=1. 

Therefore,  substituting,  a  ^  +  67  =  1, 

which  is  the  equation  desired. 

Ex.  3.     Eliminate  6  from  the  equations, 
a  cos  0  +  b  sin  #  =  c, 
d  cos  0  +  e  sin  0  =/. 

Solving  by  any  of  the  ordinary  methods  of  elimination, 

d      c<7  —  a/* 
sm0  =  —  -  J-, 
od  —  ae 


bd  —  ae 

Substituting  these  values  of  sin  0  and  cos  0  in 

sin20  +  cos20=  1, 
and  reducing,  we  have 


-        * 


(j/_  cey  +  (c,i  ._  af)2  =  (bd  -  ae) 

Ex.  4.     Eliminate  0  from  the  equations 

cot  0  +  tan  0  =  a  ;  (1) 

sec  0  —  cos  0  =  b.  (2) 

From  (1)  a  =  -1-  +  tan  0  =  1  +  taf  °. 

tan  0  tan  0 

S6C    \/  xo\ 

•••a=te^-  (3) 


174  PLANE   TRIGONOMETRY 


From  (2)  6  =  sec  0 -—  = 

sec  0          sec 


. 

sec0 

From  (3)  and  (4)  «26  =  sec3  0,  and  a&2  =  tan3  0. 

But  sec2  0-  tan2  0=1. 


(4) 


=  1, 

or,  aV  -  aV  =  1. 

Ex.  5.     Eliminate  0  from  the  equations 

-  cos  6  -  &  sin  6  =  cos  20;  (1) 

#  0 

-sin<9  +  ^cos<9:=2sin20.  (2) 

a  6 

Multiplying  (1)  by  cos  0  and  (2)  by  sin  0  and  adding  the  resulting  equa- 

tions, we  obtain  x 

-  =  cos  0  cos  20  +  2  sin  2  0  sin  0 
a 

=  cos  0  cos  2  0  +  sin  0  sin  20+  sin  0  sin  2  0 

=  cos  0  +  2  sin2  0  cos  0.  (-1) 


In  like  manner,  multiplying  (1)  by  sin  0  and  (2)  by  cos0  and  subtracting, 

=  2  sin  2  0  cos  0  -  cos  2  0  sin  0 

=  sin0+2sin0cos20.  (4) 


we  obtain  1t 

±  =  2  sin  2  0  cos  0  -  cos  2  0  sin  0 
b 


Adding  (3)  and  (4), 

-  + 1  -  cos  0  +  sin  0  +  2  sin  0  cos  0  (cos  0  +  sin  0) 
=  (cos  0  +  sin  0)  ( 1  +  2  sin  0  cos  0) 
=  (cos  0  +  sin  0)(cos2  0  +  sin2  0  +  2  sin  0  cos  0) 
=  (cos0  +  sin0)3. 

i 

(5) 

By  subtracting  (4)  from  (3)  and  reducing  the  result,  we  find  that 

(«) 


Squaring  (5)  and  (6)  and  adding  the  results,  we  obtain  the  following, 
which  is  the  desired  equation  : 


x  y 
__  ^ 
a  b 


TRIGONOMETRIC   ELIMINATION  175 

Ex.  6.     From  the  following  simultaneous  equations,  find  the 
values  of  r,  </>,  6  : 

r  sin  0cos<£  =  a;  (1) 

r  cos  0  cos  <£  =  & ;  (2) 

r  sin  <£  =  c.  (3) 

Dividing  (1)  by  (2),    ten0  =  |.     .-.  0  =  tan-*S.  (4) 

Squaring  (1)  and  (2)  and  adding, 

r2  cos2  <£  =  a2  +  62.  (5) 

Taking  the  square  root  of  (5),  and  then  dividing  (3)  by  this  result, 

c  (6) 


Va2  +  b2  Va2  +  62 

Squaring  (3) 'and  adding  the  result  to  (5), 

r2  =  a2  +  b'2  +  c2, 
r  =  Va2  +  £2  +  c2. 

EXERCISE   XXXI 

1.  Find  r  and  0  if  r  sin  0  =  1.25  and  r  cos  0  =  2.165. 

Eliminate  0  from  the  equations  following : 

2.  «  cos  0  +  6  sin  0  =  c,  and  6  cos  6  —  a  sin  0  =  c?. 

3.  -  cos  0  +  f  sin  0  =  1,  and  -  sin  0  —  ^  cos  0  =  1. 
a  b  a  b 

4.  a  sec  0—6  tan  0  =  <?,  and  c?  sec  0  +  £  tan  0=6. 

5.  a  cos  20  =  6  sin  0,  and  e  sin  2  0  =  6?  cos  0. 

6.  cos  0  +  sin  0  =  a,  and  cos  2  0  =  6. 

7.  sin  0  +  cos  0  =  a,  and  tan  0  +  cot  0=6. 

8.  cot  0  +  cos  0  =  a,  and  cot  0  —  cos  0  =  6. 

9.  sin  0  —  cos  0  =  «,  and  esc  0  —  sin  0  =  6. 

10.  sin  0  +  cos  0  sin  2  0  =  a,  and  cos  0  +  sin  0  sin  20=6. 

11.  sec  0  —  cos  0  =  a,  and  esc  0  —  sin  0  =  6. 

Eliminate  0  and  $  from  the  following  equations : 

12.  tan  0  +  tan  $  =  a,  cot  0  +  cot  ^>  =  6,  and  0  +  c£  =  a. 

13.  sin  0  +  sin  <$>  =  a,  cos  0  +  cos  $=b,  and  0  —  $  =  <*. 

14.  a  cos2  0  +  6  sin2  0  =  ccos2(£,  #sin20  +  6cos20  =  c?sin2<£, 
and  c  tan2  0  -  d  tan2  <j>  =  0. 


176 


PLANK    TRIGONOMETRY 


SPHERICAL  TRIGONOMETRY 

CHAPTER   XV 
GENERAL   THEOREMS   AND  FORMULAS 

116.  Spherical  trigonometry  is  that  branch  of  trigonometry 
which  treats  of  the  solution  of  spherical  triangles. 

117.  The  following  definitions  and  theorems  are  to  be  found 
in  works  on  solid  geometry.      For  a  discussion  of  the  defini- 
tions and  for  proofs  of  the  theorems  the  student  is  referred  to 
any  text-book  on  that  subject. 

DEFINITIONS  AND  THEOREMS 

1.  The  curve  of  intersection  of  a  plane  and  a  sphere  is  a 
circle. 

2.  A  great  circle  is  a  circle  formed  by  a  plane  that  passes 
through  the  center  of  the  sphere. 

3.  A  small  circle  is  a  circle  formed  by  a  plane  that  inter- 
sects the  sphere  without  passing  through  its  center. 

4.  Through  any  two  points  on  the  surface  of  a  sphere  one 
and  only  one  great  circle  can  be  passed,  unless  these  points 
are  at  opposite  extremities  of  a  diameter  of  the  sphere. 

5.  A  spherical  angle  is  the  angle  between  two  arcs  of  great 
circles.     It  is  equal  to  the  angle  between  the  tangents  to  the 
two  circles  drawn  at  their  point  of  intersection  ;   it  is  also  equal 
in  angular  magnitude    to    the    dihedral    angle  formed  by  the 
planes  of  the  two  great  circles. 

6.  A    spherical   polygon   is  a  portion  of  the  surface  of  the 
sphere  bounded  by  three  or  more  arcs  of  great  circles. 

7.  A  spherical  trian'gle  is  a  spherical  polygon  of  three  sides. 

CONANT'S  TRIG, —  12  177 


178  SPHERICAL   TRIGONOMETRY 

118.  Let  ABC  be  any  spherical  triangle,  and  0  the  center 
of  the  sphere  on  whose  surface  the  triangle  is  drawn.  The 

vertices  are  represented  geometri- 
cally by  the  letters  A,  B,  C,  and  the 
same  letters  are  used  to  designate 
the  angles  lying  at  these  vertices 
respectively.  The  sides  opposite 
these  angles  are  designated  by  the 
corresponding  letters  a,  b,  c.  Since 
0  is  the  center  of  the  sphere,  OA  =  OB—OC,  each  being  a 
radius  of  the  same  sphere.  Also,  the  arcs  a,  b,  c,  are  the  meas- 
ures of  the  central  angles  BOC,  AGO,  A  OB,  respectively. 

THEOREMS.  The  following  theorems  on  spherical  triangles 
were  proved  in  solid  geometry. 

I.  The  sum  of  any  two  sides  of  a  spherical  triangle  is  greater 
than  the  third  side.* 

II.  In  any  spherical  triangle  the  greatest  side  is  opposite  the 
greatest   angle,  and  conversely.     Also,  equal   sides  are  opposite 
equal  angles. 

III.  Any  angle  of  a  spherical  triangle  is  less  than  180°. 

IV.  The  sum  of  the  angles  of  a  spherical  triangle  is  greater 
than  180°  and  less  than  540°  ;  i.e.  180°  <  A  -f  B  +  0<  540°. 

V.  Any  side  of  a  spherical  triangle  is  less  than  180°. 

VI.  The  sum  of  the  sides  of  a  spherical  traingle  is  less  than 
360°  ;  i.e.  a  +  b  +  c<  360°. 

VII.  The  difference  of  any  two  angles  of  a  spherical  triangle 
has  the  same  sign  as  the  difference  of  the  corresponding  opposite 
sides  ;  e.g.  A  —  B  and  a  —  b  are  of  the  same  si</n. 

VIII.  If  from  the  vertices  of  a  spherical  triangle  as  poles,  arcs 
of  great   circles  are  drawn,  a  second  spherical  triangle  will  be 
formed  which  is  called  the  polar  of  the  first  triangle. 

Let  ABC  be  any  spherical  triangle,  and  a',  &',  c'  be  arcs  of  great  circles 
drawn  with  A,B,  C,  respectively  as  poles.  If  these  arcs  are  extended  and 
the  great  circles  are  fully  drawn,  the  surface  of  the  sphere  is  divided  into 

*  Three  great  circles  intersect  on  the  surface  of  a  sphere  in  such  a  way  as  to 
form  eight  triangles  ;  and  one  of  these  triangles  always  satisfies  the  theorems  of 
this  section.  Only  such  triangles  are  considered  in  this  work. 


GENERAL  THEOREMS  AND  FORMULAS 


179 


eight  spherical  triangles.     That  triangle  A'B'C'  is  called  the  polar  of  ABC 
which  is  so  situated  that  A  and  A'  lie  on  the  same  side  of  BC  ;  B  and  B'  on 
the  same  side  of  A  C  ;  C  and  C"  on  the  same  side  of  AB. 
»   Any  angle  of  a  spherical  triangle  is  the  supplement  of  the  side  opposite 
in  its  polar. 

HABC  is  the  polar  of  A'B'C',  then  conversely  A'B'C'  is  the  polar  of 
ABC. 

Let  ABO  arid  A'  B'  0'  be  two  polar  triangles,  and  let  a,  b,  <?, 
and  a',  6',  <?',  be  the  sides  opposite  the  like-named  angles  in  the 
two  triangles  respectively. 


A'  -180°  -a, 
B1  =  180°  -  b, 


Then,  .l  =  1800-a', 
B  =  180°  -  br, 
(7=180°-</. 

Spherical  triangles  are  called  isosceles,  equilateral,  equiangular, 
right,  and  oblique  under  the  same  conditions  as  are  the  corre- 
sponding plane  triangles. 

It  is  to  be  remembered,  however,  that  a  spherical  triangle 
may  have  one,  two,  or  three  right  angles.  If  it  contains  two 
right  angles,  it  is  called  a  bi-rectangular  spherical  triangle  ;  and 
if  it  contains  three  right  angles,  it  is  a  tri-rectangular  spherical 
triangle. 

NOTE.  The  length  of  a  side  of  a  spherical  triangle,  expressed  in  linear 
measure,  can  not  be  determined  until  the  radius  of  the  sphere  is  known. 

119.  FUNDAMENTAL  THEOREM.  To  express  a  side  of  a 
spherical  triangle  in  terms  of  the  other  two  sides  and  of  the  angle 
opposite  : 

Let  ABQ  be  a  spherical  triangle  and  0  the  center  of  the 
sphere. 


180 


SPHERICAL   TRIGONOMETRY 


E 


From  D,  any  point  in  the  radius 
OA,  draw  DE,  DF,  perpendicular 
to  OA,  in  the  planes  OAB,  OAC, 
respectively.  Connect  EF. 

Then    is   the  plane  angle  EDF 
equal   to   the  angle  A   (Art.   117, 
p.  177). 
In   the    plane   triangles    DEF,   OEF,    we     have    (Art.    96, 

P-  133)        EF2  =  DE2  +  DF2  -2DE-  DF  cos  A, 
and  EF2  =  OE2  +  OF2  -  2  OE  -  OF  cos  a. 

Equating  these  values  of  EF2  and  transposing,  we  have 

L-2  OE. 


Substituting  OD2  for  OE2  -  DE2,  and  also  for  OF2  -  DF2,  this 
becomes 

20D2  +  2DE-DFcosA-2  OE  •  OFcosa=0. 


Dividing  by  2  OE  •  OF, 

OD     OD     DE    DF 


i.e. 


cos  a  —  cos  b  cos  c  -f  sin  b  sin  c  cos  A. 


*J. 


120.  An  examination  of  the  figure  which  accompanies  the 
demonstration  in  the  preceding  article  shows  that  the  implied 
supposition  is  there  made 
that  both  b  and  c  are  less 
than  90°,  but  that  no  restric- 
tion is  placed  on  a.  ti/  \  V  \*> 

In  order  to  establish  the 
truth  of  the  theorem  for  all 
values  of  a  and  b  we  pro- 
ceed as  follows  : 

Let  b  be  greater  than  90°.     Produce  the  arcs  CA  and  OB 
until  they  intersect  again  in  C'. 

Since  AC  >  90°,  we  have  AC'  <  90°.     Therefore  in  the  tri- 
angle ABC',  A(j,      90o 

and,  by  hypothesis,  AB  <  90°, 

while  BC'  is  unrestricted. 


GENERAL  THEOREMS  AND  FORMULAS       181 

Applying  (1),  Art.  119,  to  the  triangle  ABC',  we  have 

cos  a'  =  cos  b'  cos  c  -f-  sin  br  sin  c  cos  Z  C'AB.  (1) 

But  (Art.  53,  p.  78),       cos  a'  =  —  cos  a, 

cos  br  =  —  cos  b, 
and  cos  Z  C'AB  =  —  cos  A. 

Substituting  these  values  in  (1),  we  have 

cos  a  =  cos  b  cos  c  -f-  sin  b  sin  c  cos  ^4. 

In  like  manner  it  can  be  shown  that  the  theorem  remains 
true  if  a  and  b  are  both  greater  than  90°.  Hence,  it  is  true 
for  all  spherical  triangles  which  come  within  the  scope  of 
our  work. 

Also,  by  drawing  the  perpendiculars  DE<  DF,  from  some 
point  in  the  radius  OB  in  the  planes  BOC,  BOA,  respectively, 
in  the  figure  of  Art.  119,  we  can  obtain  a  corresponding  formula 
for  expressing  the  value  of  cos  b  ;  and  by  drawing  these  per- 
pendiculars from  some  point  in  the  radius  00,  in  the  planes 
OOA,  COB,  respectively,  a  similar  formula  for  the  value  of 
cos  c.  Therefore, 

cos  a  =  cos  b  cos  c  4-  sin  b  sin  c  cos  A, 

cos  b  =  cos  c  cos  a-\-  sine  sin  a  cos  B,  (2) 

cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  C. 

The  above  are  relations  involving  the  sides  and  one  of  the  angles 
of  a  spherical  triangle. 

From  these  equations  the  following  are  at  once  derived  : 

cos  a  —  cos  b  cos  c 


cos  A  = 


T>     cos  b  —  cos  c  cos  a 
cos  -o  = 


COS 


sin  c  sin  a 

cos  c  —  cos  a  cos  b 
sin  a  sin  b 


sin  b  sin  c 

(3) 


These  relations  express  the  values  of  the  cosines  of  the  angles  of 
a  spherical  triangle  in  terms  of  the  sides  of  the  triangles. 


182  SPHERICAL   TRIGONOMETRY 

121.  After  the  first  of  the  three  formulas  in  (2)  or  in  (3) 
in  the  preceding  article  has  been  obtained  the  others  can  be 
derived  from  it  by  a  cyclic  interchange  of  the  letters  a,  £>,  c, 
replacing  at  the  same  time  A  by  B,  and  B  by  0. 

122.  The  law  of  sines.     From  plane  trigonometry  we  have 

the  relation  .  „    . 

sin2  A  =  1  —  cos2  A. 

Replacing  cos2  A  by  its  value   from   (3)   in   the  preceding 
section, 

sin2  A  =  1  —  (cos  a  ~  cos  fr  cos  g)2 
sin2  b  sin2  c 

_  sin2  b  sin2  c  —  (cos  a  —  cos  b  cos  <?)2 
sin2  b  sin2  c 


_  (1  —  cos2  #)(!  —  cos2  (?)  —  (cos  a  —  cos  b  cos  e)2 
sin2  b  sin2  c 

Expanding,  reducing,  and  rearranging  terms,  we  have 

.  2  A  _  1  —  c°s2  0  —  c°s2  &  —  cos2  0  +  2  cos  «  cos  £  cos  <? 

sin  ^i .  „  , — :— r 

snr  6  smj  c 

Dividing  both  sides  of  the  equation  by  sin2  a  and  extracting 
the  square  root,  we  obtain 

sin  A      Vl  —  cos2  a  —  cos2  b  —  cos2  c+2  cos  a  cos  6  cos  £     x-,  >. 


sin  a  sin  a  sin  o  sin  c 

In  a  precisely  similar  manner  it  can  be  proved  that  — —  and 
.       *  sin  b 

also  that  — : —  have  the  same  value.     Therefore,  since  each  of 
sin  c 

these  ratios  has  the  same  value,  they  are  equal  to  each  other. 

sin  A  _  sin  B  __  sin  0  ^^\ 

sin  a       sin  b       sin  c  ' 

which   is   the   law  of   sines.     It    may  be   stated   in  words  as 
follows : 

The  sines  of  the  sides  of  a  spherical  triangle  are  to  each  other  as 
the  sines  of  the  opposite  angles. 


GENERAL  THEOREMS  AND  FORMULAS 


183 


An  inspection  of  (1)  shows  that  a  cyclic  interchange  of  the 
letters  a,  5,  c,  and  A,  J5,  (7,  leaves  the  right  member  of  the  equa- 
tion unchanged,  while  the  left  member  is  changed  into  — —  and 
.     Q  sin  b 

— successively.     Hence,  after  (1)  has  been  proved,  (2)  can 

sin  c 

be  established  by  cyclic  interchange  of  letters. 

123.  To  derive  a  relation  involving  the  angles  and  one  of  the 
sides  of  a  spherical  triangle. 

Let  A'JS1 '0'  and  ABC  be  two  spherical  triangles  polar  to  each 
other.  Then  (Art,  118,  p.  1T9), 

a'  =  180°  -  A, 

bf  =  180°  -  B, 

c'  =  180°  -  0. 
By  (1),  Art.  119,  p.  180, 

cos  a'  =  cos  b'  cos  c'  +  sin  bf  sin  c'cos  A! . 
But,  by  (1), 


cos  a'  =  —  cos  A, 
cos  b'  =  —  cos  B, 
cos  c1  =  —  cos  C. 


sin  b1    =  sin  _Z?, 

sin  c'    =  sin  (7, 
cos  A'  =  —  cos  a. 


Substituting  these  values  in  (2),  we  have 

—  cos  A  =  cos  B  cos  C—  sin  .Z?  sin  (7 cos  a. 

In  like  manner  we  can  obtain  corresponding  values  for  cos  B 
and  for  cos  0. 
Therefore, 

cos  A  =  —  cos  B  cos  (7+  sin  B  sin  C  cos 


cos  B  =  —  cos  (7  cos  ^4  +  sin  (7 sin  A  cos  6, 
cos  0  =  —  cos  J.  cos  B  +  sin  .A  sin  B  cos  <?. 

From  these  equations  the  following  are  at  once  derived 
cos  A  +  cos  B  cos  C 


(3) 


cos  a  = 


cos  b 


cose 


sin  J5  sin  0 

cos  B+  cos  (7  cos  J. 

sin  C  sin  ^1 

cos  (7  +  cos  A  cos  .g 
sin  A  sin  .B 


(4) 


184  SPHERICAL   TRIGONOMETRY 

124.   To  derive  a  relation  involving  two  angles  and  the  sides 
of  a  spherical  triangle. 

Resuming  (1),  Art.  119,  p.  180,  we  have 

cos  a  =  cos  b  cos  e  +  sin  b  sin  c  cos  A. 

Substituting  in  this  equation  the  value  of    cos  c  obtained 
from  (2),  Art.  120,  p.   181, 

cos  a  =  cos  b  (cos  a  cos  b  -f  sin  a  ski  b  cos  (7)  -f-  sin  b  sin  c  cos  A 
=  cos  a  cos2  b  +  sin  a  sin  £>  cos  b  cos  (7  +  sin  b  sin  <?  cos  A. 
cos  a  (1  —  cos2  6)  =  sin  a  sin  6  cos  b  cos  (7  4-  sin  b  sin  <?  cos  A. 

Substituting  for  1  —  cos2  b  its  value,  sin2  6,  and  dividing  both 
sides  of  the  equation  by  sin  6,  we  obtain  the  desired  relation, 

cos  a  sin  b  =  sin  a  cos  b  cos  C  +  sin  c  cos  A.  (1) 

In  like  manner  we  can  obtain  corresponding  expressions  for 
the  value  of  cos  a  sin  c,  of  cos  b  sin  c?,  etc.     Therefore, 

cos  a  sin  b  =  sin  a  cos  b  cos  (7  +  sin  c  cos  .A, " 
cos  a  sin  c  =  sin  a  cos  c  cos  5  -f  sin  b  cos  A, 
cos  5  sin  a  =  sin  6  cos  a  cos  (7  +  sin  c  cos  .#, 
cos  b  sin  <?  =  sin  b  cos  <?  cos  A  4-  sin  a  cos  .B, 
cos  c  sin  6  =  sin  c  cos  5  cos  A  +  sin  a  cos  (7, 
cos  c  sin  #  =  sin  c  cos  a  cos  B  -f-  sin  5  cos  (7,  . 


(2) 


125.   To  derive  a  relation  involving  two  sides  and  the  angles 
of  a  spherical  triangle. 

Resuming  the  first  equation  under  (3),  Art.  123,  p.  183,  we 
cos  A—  —  cos  B  cos  C  +  sin  B  sin  0  cos  a. 

Substituting  in  this  equation  the  value  of  cos  C  obtained 
from  the  third  equation  of  the  same  set, 

cos  A  =  —  cos  B  (  —  cos  A  cos  B  -f  sin  A  sin  B  cos  <?) 

4-  sin  B  sin  C  cos  a 

=  cos  A  cos2  J5  —  sin  A  sin  i?  cos  B  cos  c  4-  sin  B  sin  (7  cos  a. 
Transposing  and  factoring, 
cos  A  (1  —  cos2  B)  =  —  sin  A  sin  B  cos  B  cos  6>  +  vsin  B  sin  (7  cos  a. 


GENERAL  THEOREMS  AND  FORMULAS 


185 


Replacing  1  —  cos2  B  by  its  value,  sin2  B,  and  dividing  both 
sides  of  the  equation  by  sin  B,  we  obtain  the  desired  relation, 

cos  A  sin  B  —  cos  a  sin  0  —  cos  c  sin  A  cos  B.  (1) 

In  like  manner  we  can  obtain  corresponding  expressions  for 
the  value  of  cos  A  sin,  (7,  cos  B  sin  J.,  etc.     Therefore, 

cos  A  sin  1?  =  cos  a  sin  (7  —  cos  c  cos  ^  sin  A, 
cos  A  sin  (7  =  cos  a  sin  .5  —  cos  b  cos  (7  sin  A, 
cos  (7  sin  5  =  cos  c  sin  A.  —  cos  a  cos  jB  sin  (7, 
cos  (7  sin  A  =  cos  <?  sin  B  —  cos  5  cos  A  sin  (7, 
cos  J5  sin  A  =  cos  6  sin  0  —  cos  <?  cos  A  sin  5, 
cos  B  sin  (7  =  cos  6  sin  A.  —  cos  a  cos  (7  sin  ^. 


(2) 


126.    From  the  formulas  in  Art.  124  a  group  of  important 
relations  is  derived,  as  follows : 

From  the  first  of  the  six  formulas  there  given  we  have 

cos  a  sin  b  =  sin  a  cos  b  cos  (7  +  sin  c  cos  A. 
Dividing  both  sides  of  the  equation  by  sin  a, 


sin 


sin  a 


T>     T     .       sin  c  ,  ,  sin  C   .-,  •    i 

Replacing  —    -  by  its  equal -,  tins  becomes 

sin  a  sin -4 

cot  a  sin  b  =  cos  b  cos  (7+  sin  C • 

sin  A 

.  *.  cot  a  sill  b  =  cos  5  cos  C  -f-  sin  (7  cot  A. 


In  like  manner  we  can  obtain  corresponding  expressions  for 
the  value  of  cot  a  sin  <?,  cot  b  sin  #,  etc.     Therefore, 

cot  a  sin  b  =  cos  5  cos  (7+  sin  (7 cot  A., 
cot  a  sin  c  —  cos  c  cos  B  +  sin  ^  cot  A, 

cot  5  sin  c  =  cos  <?  cos  A  -\-  sin  -A  cot  B, 

r  (^) 

cot  5  sin  a  =  cos  a  cos  (7  +  sin  (7  cot  B, 

cot  c  sin  a  =  cos  a  cos  5  +  sin  B  cot  (7, 
cot  c  sin  5  =  cos  b  cos  .A  +  sin  A  cot  (7. 


186  SPHERICAL   TRIGONOMETRY 

127.   The  values  of  sin  ^,  cos  ^,  tan  -,  etc.,  in  terms  of  the 
sides  of  the  triangle. 

From  (3),  Art.  120,  p.  181, 


cos  A  = 


cos  a  —  cos  b  cos 


sin  b  sin  c 
From  this  we  have 

1  ,    A  —  S^n  ^  s^n  C  +  COS  ^  cos  c  —  cos 

X  —  COo  */l  — 


sin  b  sin  c 

by  Art.  68,  =  cos  (b  ~  c) .~  cos  a . 

sin  b  sin  c 

Dividing  by  2,  and  applying  (8),  Art.  77,  p.  100, 

.     a-4-  b  —  c    .    a—  b  4-  c 
sin  -  -  sin  - 

1  —  cos  A  2 


sin  b 


sn  6- 


Putting  #-f5-h&  =  2s,  and  replacing  -      ^()S       by  its  equal 


-   %A_  sin  (g  —  5)  sin  (>  —  c) 
2  sin  6  sin  c 


.  ..  sn      = 


2  sin  6  sin  c 

In  like  manner, 

1   .  j  _  sin  b  sin  c  —  cos  b  cos  <?  +  cos  a 

\.  -f-  COS  J\.  —  --  — 

sin  b  sin  c 

1  +  cos  A  _  cos  a  —  cos  (6  +  c) 

2  2  sin  b  sin  £ 


S  =  _  2  _  : 

2  sin  6  sin 


cos     =  -  .  (2) 

2       *       sin  i  sin  c 


GENERAL  THEOREMS  AND  FORMULAS       187 

Dividing  (1)  by  (2),  we  have 


tan     = 


2       *      sin  8  sin  (s  —  a) 

Since  any  angle  of  a  spherical  triangle  is  less  than  180°,  all 

the  functions    of   the   half   angles   are    positive;    i.e.    sin    —  , 

A          A  ^ 

cos—,  tan  —  ,  are  all   positive.     Therefore  the    signs   of   the 

'2.  A 

radical  expressions  in  (1),  (2),  and  (3)  are  positive. 

Since  s,  a,  5,  c,  s  —  a,  s  —  b,  s  —  c  are  severally  less  than  180° 
and  positive,  the  values  obtained  in  (1),  (2),  and  (3)  are  real. 

128.   The  values  of  sin  |,  cos  ^,  tan  |,  etc.,  in  terms  of  the 
angles  of  the  triangle. 

From  (4),  Art.  123,  p.  183, 

cos  A  4-  cos  B  cos  O 


cos  a  = 


Therefore,  1  —  cos  a  = 


sin  £  sin  C 

sin  B  sin  C '  —  cos  B  cos  0  —  cos  A 
sin  B  sin  C 


—  cos  (B  +  (7)  —  cos  A 
sin  B  sin  C 

-,  i   .  sin  B  sin  (7  -f  cos  5  cos  (7+  cos  A 

and  1  +  cos  a  =  — 

sin  B  sin  (7 

_  cos  (B  —  O)  4-  cos  A 
sin  .2?  sin  (7 

Putting  A  +  B  -f-  0=2  S,  and  proceeding  as  in  the  last  sec- 
tion, we  obtain 

.     a        l—GOStScoafS— A) 
-      sin  J  sing          ' 


sin  5  sin  C 


—  cos  iS  cos  T/S7  — 


188  SPHERICAL   TRIGONOMETRY 

Since  any  side  of  a  spherical  triangle  is  less  than  180°,  all 
the  functions  of  the  half  sides  are  positive;  i.e.  sin  ^,  cos-, 

tan  -,   are  all   positive.     Therefore  the   signs  of   the    radical 

2 

expressions  in  (1),  (2),  and  (3)  are  positive. 

To  prove  that  these  expressions  are  real  we  proceed  as  follows  : 
Let  A'B'C'  be  the  polar  triangle  of  ABC,  and  let  a',  b1,  c', 

be  the  sides  of  A'B'  0'  which  lie  opposite  the  angles  A,  B,  C, 

respectively  of  the  original  triangle. 

Then,  since  a',  6',  <?',  are  supplements  of  A,  B,  (7,  respectively, 

and  since  a1  <  b'  +  e',  we  have 

180°  -A<  (180°  -  B)  +  (180°  -  (7). 
Transposing,  B  +  C  -  A  <  180° ; 

i.e.  S-A<90°. 

Therefore,  cos  (S—  A)  is  positive. 

Also,  since  A  +  B  +  0  lies  between  360°  and  540°,  8  lies 
between  180°  and  270°.  Hence  cos  8  is  negative  ;  i.e.  —  cos  8 
is  positive. 

Therefore  the  radical  expressions  in  (1),  (2),  and  (3)  are 
real. 

129.   Gauss's  equations.     From  Art.  69,  p.  92, 

(A  ,  B\  A        B  A       B 

C°S  U      ~2  J  =  C°S  2"  C°S  "2  ~  S1U  "2  Sm  "2  ' 

A  A 

Substituting  in  this  equation  the  values  of  cos  —  and  sin  — , 

T)  T> 

and    corresponding   values    for   cos--    and   sin-'    (Art.    127, 
p.  186),  we  have 


cos  fA+.lP\m     /sin  g  sin  Q  -  a}  ^      /sin  *  sin  ( 
\2       2y       ^       sin  b  sin  c  ^      sin  a  si 


8-5) 

Sill  0 


/sin  (8  —  b)  sin  (s  —  <?)        /sin  (s  —  #)  sin  (s  — 
^  sin  b  sin  <?  ^  sin  a  sin  <? 


sin  s  —  sin  ( s  — 
sin 


sin  (s  —  c)  A/sin  (s  —  a)  sin  (s  —  5) 
in  c  *  sin  6  sin  a 


GENERAL  THEOREMS  AND  FORMULAS       189 

But  by  Art.  77,  p.  100,  and  Art.  80,  p.  106, 

o        2  s  —  c    .    c 
2  cos  — - —  sm  - 


sin  s  —  sin  (s  —  <?)  __  2  2 

!smj 

a  +  b 


sin  c  n    .     c         c 

2  sm  -  cos  - 


cos 


COS  9 

and  by  Art.  127,  p.  186, 

/sin  (s  —  a)  sin  (s  —  b)  _     .     O 
*  sin  a  sin  b  2 

Substituting  these  values  in  (1),  and  reducing,  we  have 


A+B  o 

008  -3-  =  —  r~sm2'  (2) 

cos- 

Tn  like  manner  corresponding  values  can  be  obtained  for 
sin  —  -  —  ,  sin  —  ^  —  ,  and  cos  —  —  —  .  These  four  relations, 
which  are  commonly  known  as  Gauss's  Equations,  are  as  follows  : 

a  +  b 

A  +  B  ~2~    .     0 

cos  —^~-  =  -  —  sin  -g-  ;  (3) 

cos- 

a-b 

A  +  B     COS^~         O 
sm  —  g—  =  -  —  cos  ^-5  (4) 

cos- 
.    a  +  b 


sn 


.    a-b 
sin  - 
.    A-B  2  C 

sm—  -  —  =  -  T~cos"o' 
sin- 


190  SPHERICAL   TRIGONOMETRY 

130.  Napier's  analogies.  From  Gauss's  Equations  the  follow- 
ing are  derived.  The  method  of  derivation  is  obvious,  and  the 
work  is  left  as  an  exercise  for  the  student. 

a  —  b 
cos  —  -  — 


COS 


(3) 


sin 


131.  Special  formulas  for  the  solution  of  spherical  right  tri- 
angles. If  one  of  the  angles  of  the  triangle,  as  (7,  is  a  right 
angle,  the  following  special  formulas  are  derived  from  those 
established  in  the  preceding  sections  : 

From  (2),  Art.  120,  p.  181, 

cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  C.  (1) 

But,  since  (7=  90°,  cos  0=  0.     Therefore  the  second  term  of 
the  right  member  becomes  zero.     Therefore, 
cos  c  =  cos  a  cos  b. 

In  a  manner  similar  to  that  just  employed,  the  following 
formulas  are  derived  for  the  special  case  when  C  is  a  right  angle. 

From  (2),  Art.  122,  p.  182,  formulas  for  finding  either  of  the 
oblique  angles  when  the  hypotenuse  and  the  opposite  leg  are 

Siven-  sin  a- 

sin  A  =  -  —  , 
sin  c 

j  -     T>      sin  b 

arid  sin  B—  -   — 

sin  c 


GENERAL  THEOREMS  AND  FORMULAS       191 

From  (3),  Art.  123,  p.  183,  formulas  for  finding  either  of 
the  oblique  angles  when  the  opposite  leg  and  the  other  oblique 

angle  are  given. 

cos  A  =  cos  a  sin  B,  1 

cos  B  =  cos  b  sin  A.  J 

From  (2)  and  (3)  are  derived   the  following  formulas  for 
finding  an  oblique  angle  when  the  hypotenuse  and  the  adjacent 

leg  are  given. 

cos  A  =  tan  b  cot  <?, 

cos  B  =  tan  a  cot  c. 


From  (2),  Art.  126,  p.  185,  formulas  for  finding  the  oblique 
angles  when  the  legs  are  given. 


,,      tan  a 

tan  A  = •, 

sin  b 


tan  B  = 


tan  b 


sin  a 


(5) 


From  (3),  Art.  123,  p.   183,  formulas  for  finding  the  legs 
when  the  two  oblique  angles  are  given. 


cos 


, 

sin  B 


sin  A 

Multiplying  together  the  two  formulas  just  obtained,  and 
replacing  the  left  member  of  the  product,  cos  a  cos  5,  by  its 
value  given  in  (1),  we  have  the  following  formula  for  finding 
the  hypotenuse  when  the  two  oblique  angles  are  given  : 

cos  c=  cot  A  cot  B.  (7) 

132.  Napier's  rules.  The  formulas  of  the  last  section  are 
sufficient  for  the  solution  of  every  possible  case  that  can  arise 
under  spherical  right  triangles.  But  it  is  often  better  to  solve 
the  various  cases  that  arise  under  right  triangles  by  two  con- 
venient and  simple  rules  devised  by  Napier,  the  inventor  of 
logarithms. 


192  SPHERICAL   TRIGONOMETRY 

These  rules  are  constructed  by  supposing  that  a  right  tri- 
angle has  five  parts.  These  parts,  which  are  usually  called 
Napier's  parts,  are 

(1)  The  two  legs. 

(2)  The  complement  of  the  hypotenuse. 

(3)  The  complements  of  the  two  oblique  angles. 

The  right  angle  is  not  considered,  and  plays  no  part  what- 
ever in  the  solution  of  a  triangle  by  this  method. 

Any  one  of  the  five  parts  may  be  regarded  as  the  middle 
part.  The  two  parts  immediately  adjacent  to  this  are  called 
the  adjacent  parts,  and  the  other  two  are  called  the  opposite 
parts. 

Napier's  rules  for  the  solution  of  spherical  right  triangles 
are  as  follows  : 

1.  The  sine  of  the  middle  part  is  equal  to  the  product  of  the 
tangents  of  the  adjacent  parts. 

2.  The  sine  of  the  middle  part  is  equal  to  the  product  of 
the  cosines  of  the  opposite  parts. 

The  similarity  of  the  vowel  sounds  in  the  syllables  tan-,  ad- 
and  co-,  op-  renders  it  easy  to  remember  these  rules,  and  also 
to  distinguish  them  from  each  other. 

To  test  the  correctness  of  these  rules,  assume  any  three  parts 
as  the  given  parts.  For  example,  let  the  given  parts  be  a,  b, 

and  co-A.  In  this  case  b  is  the  middle 
part,  and  a,  co-A,  are  to  be  considered 
adjacent  parts.  Hence  we  have 

sin  b  =  tan  a  tan  (co-A) 
=  tan  a  cot  A. 

This  is  the  same  as  the  first  of  the  two 
formulas  under  (5),  Art.  131,  p.  191, 
which  has  already  been  proved  to  be 
true. 

As  another  illustration,  let  the  given  parts  be  a,  co  A,  co-B. 
Here  co-A  is  the  middle  part,  and  a,  co-B  are  to  be  considered 
opposite  parts.  Hence 

sin  (co-A)  =  cos  a  cos  (co-B), 
cos  A  —  cos  a  sin  B. 


GENERAL  THEOREMS  AND  FORMULAS       193 

Tliis  is  the  same  as  the  first  of  the  two  formulas  under  (3), 
Art;  130,  p.  190,  which  has  already  been  proved  to  be  true. 

In  like  manner  Napier's  rules  as  applied  to  any  other  group 
of  three  parts  will  be  found  to  reduce  to  one  of  the  formulas 
already  proved.  ^ 

133.  DEFINITION.  Two  angles,  or  an  angle  and  a  side,  are 
said  to  be  of  the  same  species  when  both  are  greater  or  both  are 
less  than  90°;  they  are  said  to  be  of  opposite  species  when 
one  is  greater  and  the  other  is  less  than  90°. 

In  any  right  triangle  if  a  and  b  are  of  the  same  species,  the 
hypotenuse  c  is  less  than  90°  ;  if  a  and  b  are  of  opposite  species, 
c  is  greater  than  90°. 

This  follows  from  (1),  Art.  130,  p.  190.  For  if  a  and  b  are 
both  greater  or  both  less  than  90°,  the  product  cos  a  cos  b  is  posi- 
tive. Therefore  cos  c  is  positive  ;  therefore  c  is  less  than  90°. 

But  if  a  and  b  are  of  opposite  species,  the  product  cos  a 
cos  b  is  negative.  Therefore  cos  c  is  negative  ;  therefore  c  is 
greater  than  90°. 

EXERCISE  XXXII 

1.  Prove  that  in  any  right  triangle  a  leg  and  the  angle  oppo- 
site are  of  the  same  species. 

2.  By  the  aid  of  Napier's  rules  derive  the  formulas  in  (6), 
Art.  131,  p.  190. 

3.  If  the  hypotenuse  of  a  right  triangle  is  equal  to  90°,  what 
must  be  the  values  of  a  and  b?     Why? 


4.    Prove  tan2      = 


2      sin  (e  -f-  a) 
5.    Prove 


2      cos(B-A) 

6.  If  a  =  90°  and  b  =  90°,  what  must  be  the  values  of  the 
remaining  parts  of  the  right  triangle? 

7.  In  a  right  triangle  a  side  and  the  hypotenuse  are  of  the 
same  or  of  opposite  species  according  as  the  included  angle  is 
less  or  greater  than  90°. 

CONANT'S  TRIO.  —  13 


CHAPTER   XVI 
SOLUTION   OF    SPHERICAL    TRIANGLES 

134.  A  spherical  triangle  is  determined  when  any  three  of 
its  parts  are  known.     That  is,  when  any  three  parts  are  given, 
the  remaining  parts  can  be  computed. 

In  the  solution  of  spherical  triangles  we  have  six  cases  to 
consider,  as  follows :  having  given, 

(1)  The  three  sides. 

(2)  Two  sides  and  the  included  angle. 

(3)  Two  sides  and  the  angle  opposite  one  of  them. 

(4)  Two  angles  and  the  side  opposite  one  of  them. 

(5)  Two  angles  and  the  included  side. 

(6)  The  three  angles. 

135.  The  right  triangle.     We  proceed  first  to  the  considera- 
t  ion  of  the  right  triangle.    We  shall  suppose  that  0  is  the  right 
angle ;  and  here,  as  in  Plane  Trigonometry,  only  two  parts  are 
known  in  addition  to  the  right  angle. 

136.  Ambiguous  cases.     Whenever  a  solution  is  obtained  by 
means  of  the  sine  or  the  cosecant,  the  solution  is  ambiguous, 
because,  both  sine  and  cosecant  being  positive  in  the  second 
quadrant  as  well  as  in  the  first,  a  given  value  of  either  of  these 
functions  is,  in  general,  satisfied  by  two  angles,  one  in  the  first 
and  the  other  in  the  second  quadrant. 

Hence,  whenever  a  required  part  of  a  spherical  triangle  is 
found  by  means  of  the  sine  or  the  cosecant,  it  is  necessary  to 
test  the  result,  and  to  determine  whether  or  not  both  solutions 
are  admissible. 

When  a  solution  is  found  by  means  of  the  cosine,  tangent, 
cotangent,  or  secant,  there  is  no  ambiguity,  since  each  of  these 
functions  is  positive  in  the  first  quadrant  and  negative  in  the 
second  quadrant. 

194 


SOLUTION   OF   SPHERICAL   TRIANGLES  195 

For  this  reason  it  is  of  great  importance  that  the  student 
should  note  carefully  the  sign  of  each  of  the  functions  that 
appear  in  an  equation. 

137.   CASE  1.     Given  two  legs,  a  and  6;  to  find    c,  A,  B. 

The  formulas  for  solution  are  contained  in  (1)  and  (5),  Art. 
131,  p.  190,  or  are  obtained  directly  from  Napier's  Rules,  and 
are  as  follows  : 

cos  c  =  cos  a  cos  b  ;  (1) 

(2) 


tan  5-^.  (3) 

sm  a 

For  a  check  formula  use  cos  c  =  cot  A  cot  B. 

Ex.  1.    Given  a  =  46°  50',  b  =  31°  15';  find  c,  A,  B. 

log  cos  a  =  9.83513  -  10 
log  cos  b  =  9.93192  -  10 
log  cos  c  =  976705  10.  ]og  ^  ,  =  „  ^  _  w 

log  sin  a  =  9.86295  -  10 

log  tan  a  =  10.02781  -  10  loS  tan  B  =  9'9o20U  ~  10- 

colog  sin  b  =  10.28502  -  10  B  ~  39°  45'  32"' 

log  tan  A  =  10.31283-  10. 
A  =  64°  3'  9". 

Since  c  is  obtained  by  means  of  its  cosine  and  A  and  B  by 
means  of  their  tangents,  there  is  no  ambiguity  respecting  the 
result.  Both  a  and  b  are  in  the  first  quadrant  ;  therefore  cos  a 
and  cos  b  are  positive.  It  follows  from  this  that  the  right  mem- 
ber of  (1)  is  positive  when  applied  to  this  particular  problem  ; 
therefore  cos  c  is  positive,  and  consequently  c  is  in  the  first 
quadrant. 

In  like  manner  it  can  be  shown  that  A  and  B  are  in  the  first 
quadrant. 

When  only  one  solution  exists  that  will  satisfy  the  conditions 
of  a  problem,  the  solution  is  said  to  be  unique. 

Ex.  2.    Given  a  =  38°  44'  40",  b  =  42°  26'  28"  ; 

find  c  =  54°  51'  37",  A  =  49°  56'  12",  B  =  55°  36'  44". 


196  SPHERICAL   TRIGONOMETRY 

138.  CASE  2.  Given  the  hypotenuse  c,  and  one  of  the  legs 
a;  to  find  b,  A,  B.  The  formulas  for  solution  are  (Art.  131, 
p.  190) 

cos  5  = 


sin  A  = 


cos  a 

sin  a 

sin  c 


n     tan  a 

cosj5  = . 

tanc 

For  a  check  formula  use 

cos  B  =  cos  b  sin  A  (Art.  131,  p.  191). 

The  solutions  for  b  and  B,  being  obtained  in  each  case  by 
means  of  a  cosine,  are  unique. 

The  solution  for  A,  being  obtained  by  means  of  its  sine,  is 
apparently  ambiguous.  But  by  Art.  133,  p.  193,  a  and  A  are 
of  the  same  species.  Hence,  as  a  is  given,  the  species  of  A 
becomes  known  at  once,  and  the  ambiguity  disappears. 

Ex.1.   Given  c  =  54°  36'  30",    a  =  23°  17'  40"; 

find      b  =  50°  54'  30",  A  =  29°    1'    5",    £=72°  11'  20". 

Ex.  2.  Given  c  =  98°  15'  12",    a  =  133°  40'  24" ; 

find      b  =  78°    0'    7",  A  =  133°    2'  30",  B  =  81°  15'  40". 

139.  CASE  3.  Given  one  of  the  legs  a  and  the  opposite  angle 
A;  to  find  ft,  c,  B.  The  formulas  for  solution  are  as  follows, 

(Art.  131,  p.  190): 

sm  a 
sin    c  = 


sin   b  = 


sm  ±f  = 


sin  A* 

tan   a 
tan  A 

cos  ^4. 

cos  a 


For  a  check  formula  use     sin    b  =        -^  .        (Art.  131,  p.  191) 

tan  A 


SOLUTION    OF   SPHERICAL   TRIANGLES  197 

The  solution  is  ambiguous,  being  obtained  in  each  case  by 
means  of  a  sine.  The  different  cases  that  may  arise  are  as 
follows  : 

(1)  If  a  =  A,  then  sin  a  =  sin  A,  tan  a  =  tan  A,  and  cos  a  =  cos 
A  i   therefore  sin  <?  =  !,  sin  6  =  1,  and  sin  .5=1.     Hence  the 
solution  is  unique. 

(2)  If  c  and  a  are  of  the  same  species,  then  B  <  90°  ;  there- 
fore b  <  90°  (Ex.  7,  p.  193). 

(3)  If  c  and  a  are  of  opposite  species,  then  B  >  90°;  there- 
fore b  >  90°  (Ex.  7,  p.  193). 

After  c  has  been  computed  b  and  B  may  be  found,  if  other 
formulas  than  those  given  above  are  desired,  by  the  following 
(Art.  131,  p.  191): 


cos    b  = 


COS   Jt5  = 


cos  a 

tan  a 
tan  c 


These  formulas  give  unique  solutions  for  b  and  B,  but  for 
obtaining  <?  it  is  necessary  to  make  use  -of  the  sine.  As  any 
given  value  of  the  sine  is  satisfied  by  two  supplementary  values 
of  the  angle,  this  case  often  gives  two  solutions. 

Ex.1.   Given  a=   70°  55'  50",  ^=82°  58'    6"; 

Find      Cj  =    72°  13'  45",      ^  =  20°  54'  18",  Bl  =  22°  0'  19". 
or,  c2  =  107°  46'  15",      b2  =  159°  5'  42",  B2  =  157°  59'  41". 

Ex.  2.  Given  a=  76°  59'  59",  JL  =  39°  50'  56". 
The  triangle  is  impossible.     Why  V 

140.  CASE  4.  Given  one  of  the  legs  a  and  the  adjacent 
angle  B,  to  find  c9  6,  A.  The  formulas  for  solution  are 

(Art.  131,  p.  191)  _  tan  a 

tun    c — j — , 
cos  B 

cos  A  =  cos  a  sin  B, 
tan   b  =  sin  a  tan  B. 

For  a  check  formula  use  7 

cos  A  =  !=££;         (Art.  131,  p.  191) 
tan  c 

The  solution  is  unique.     Why? 


198  SPHERICAL   TRIGONOMETRY 

Ex.  l.    Given  a  =  21°  5'  15",   B  =  39°  8'  10"  ; 

find       c  =  26°  26'  6",    A  =  53°  55'  13",  b  =  16°  19'  5". 

Ex.  2.   Given  a  =  59°  27'  32",  B  =  36°  24'  25"  ; 

find      c  =  64°  35'  56",    b  =  32°  25'  17",  A  =  72°  26'  47". 

141.  CASE  5.    Given  the  hypotenuse  c  and  one  of  the  oblique 
angles   A;   to   find   a,  b,  B.      The   formulas   for   solution   are 

(Art.  131,  p.  190)       . 

sin  a  =  sin  c  sin  A, 

tan  b  =  tan  c  cos  A, 
cot  .Z?  =  cos  c  tan  .A. 

For  a  check  formula  use 

sin  a  =  tan  b  cot  .B  (Art.  131,  p.  191). 

The  solution  for  a,  being  obtained  by  means  of  its  sine,  is 
apparently  ambiguous.  But  since  A  is  given,  and  since  a  and 
A  are  of  the  same  species,  the  proper  value  of  a  can  always  be 
determined.  Hence  the  solution  is  unique. 

Ex.  i.   Given  c=  117°  39'  48",  A  -127°  20'  25"; 

find    a  =  135°  14'  18",  b  =  49°  9'  58",     B  =  58°  40'  37". 

Ex.2.   Given  e  =  68°  50'  31",    4  =  55°  11'  17"; 

find     a=49°58',  5  =  55°  51'  53",  5  =  62°  33'  58". 

142.  CASE  6.    Given  the  two  oblique  angles  A,  B;  to  find 
«,  b,  c.     The  formulas  for  solution  are  (Art.  131,  p.  191) 


sin 

7      cos  B 

cos  b  =  -  - 

sin  A"1 
cos  c  =  cot  A  cot  B. 

For  a  check  formula  use 

cos  c  =  cos  a  cos  b  (Art.  131,  p.  190). 
The  solution  is  unique. 


SOLUTION  OF   SPHERICAL   TRIANGLES  199 

Ex.l.   Given  4  =  63°  25' 32",  5  =  136°  1' 27"; 

find      a  =  49°53'16",     b  =  143°  34' 30",  c=  121°  13' 34". 

Ex.  2.   Given  A  =  119°  20'  !!",.#  =  114°  V  35"; 

find      a=122°28'6",    £  =  117°  57' 42",  <?  =  75°  25' 16". 

143.  The  isosceles  spherical  triangle.  An  isosceles  spherical 
triangle  can  always  be  solved  by  means  of  the  formulas  em- 
ployed in  the  solution  of  spherical  right  triangles  ;  for,  by  pass- 
ing an  arc  of  a  great  circle  through  the  vertex  and  the  middle 
point  of  the  side  opposite,  the  isosceles  triangle  can  always  be 
divided  into  two  symmetrical  right  triangles. 

EXERCISE  XXXIII 

1.  In   a    right    spherical    triangle   given   c  =  20°   50'   52", 
a  =15°  12'  44";  find  6,  A,  B. 

2.  In   a   right   spherical   triangle    given    a  =  75°   28'   24", 
b  =  33°  37'  8"  ;  find  c,  A,  B. 

3.  In    a    right    spherical    triangle    given   a  =  66°    9'    9", 
^  =  155°  49' 46";   find  b.  c,  B. 

4.  In    a    right     spherical     triangle     given     a  =  122°    5', 
B  =  125°  40' ;  find  5,  c,  A. 

5.  Iii   a   right   spherical   triangle    given    <?  =  115°   35'   4", 
J.  =  57°29';  find  a,  b,  B. 

6.  In   a   right   spherical    triangle    given    A  =  45°   23'   8", 
B  =  58°  17' ;  find  a,  b,  e. 

7.  In   a   right    spherical    triangle   given   c  =  80°   28'   44", 
A  =  33°  20'  24" ;  find  a,  6,  B. 

8.  In    a    right    spherical     triangle     given    e  =  139°    42', 
a  =  21°  47'  46" ;  find  6,  A,  B. 

9.  In    a    right    spherical    triangle    given     a  ="110°    38', 
B  =  153°  55'  40" ;   find  5,  c,  A. 

10.  In  a  right  spherical  triangle  given  a  =  112°  49', 
^  =  100°  27';  find  5,  c,  B. 


200  SPHERICAL   TRIGONOMETRY 

11.  In     a     right     spherical     triangle     given     a  =  55°    52' 
b  =  34°  46'  42"  ;  find  <?,  A,  B. 

12.  In    a    right    spherical     triangle     given     A  —  54°     20', 
^=64°  49'  51";  find  a,  b,  c. 

13.  In    a    right    spherical    triangle    if    a  =6,    prove    that 
cos2  a  =  cos  c. 

14.  In  a  right  spherical  triangle  prove  that 

sin  b  =  cos  c  tan  a  tan  B. 

15.  In  a  right  spherical  triangle  prove  that 

sin2  A  +  sin2  B  =  1  +  sin2  a  sin2  ^. 

16.  In  a  right  spherical  triangle  prove  that 

.  sin  (5  +  c)  =  2  cos2  —  cos  5  sin  <?. 


THE   OBLIQUE   SPHERICAL   TRIANGLE 

144.    In  solving  oblique  spherical  triangles  we  have  six  cases 
to  consider,  as  follows  : 

CASE  1.     Given  the  three  sides  «,  &,  c;  to  find  A,  B9  c. 

The  formulas  for  solution  are  (Art.  127,  p.  186) 


sin  g  sin  («  —  a) 


tan      = 


2  sin  s  sin  («  -  5) 

tan  ^=  Jain  («-  a)  sin  0-i)^ 

2  sin  s  sin  (s  —  (?) 

The  corresponding  formulas  for  the  sines  or  for  the  cosines 
of  the  half  angles  may  be  employed  (Art.  127,  p.  186),  but  in 
general  the  tangent  formulas  are  to  be  preferred. 

If  all  the  angles  are  to  be  found,  a  saving  of  labor  can  be 
effected  in  the  following  manner. 


SOLUTION   OF   SPHERICAL   TRIANGLES  201 

Multiply  both  numerator  and  denominator  of  the  fraction 
under  the  radical  sign  in  (1)  by  sin  (s  —  a).     Then  let 


tan  r  =  -Jsin  ( *  -  a)  sin  (s  -  6 )  sin  Q  -  c) 


sin  s 


and  we  may  write 


2      sin  (s  —  a) 

Making  the  corresponding  changes  in  (2)  and  (3),  we  have 
the  three  equations : 


tan     = 


2      sin  (s  —  a) ' 


tanr 


2      sin    «-fi' 


(7          tan  r 
tan  —  = 


2      sin  (s  —  c') 

If  these  formulas  are  employed,  it  will  be  found  that  the 
work  of  solution  can  be  more  compactly  arranged  and  more 
conveniently  carried  out  than  by  the  use  of  any  other  method. 

Ex.  1.    Given  a  =  51°  43'  18",  b  =  38°  2'  20",  c  =  75°  11'  30"  ; 

find  A. 

a  =  51°  43'  18"  log  sin  (s  -b)  =  9.84518  -  10 

6  =  38°    2'  20"  log  sin  («-c)  =  9.10311  -10 

c  =  75°  11'  30"  colog  sin  s  =  0.00375 

2  s  =  164°  57'  8"  colog  sin  (s-a)=  0.29127 

•  =  *rwu»  ^331-20 

s-a  =  30°  45'  16"  Jog  ^nA=        ^IQQ  -  10 

s  -  b  =  44°  26'  14" 


s  -  c  =    7°  17'    4"  4.=  22°  42'  27.4 


s  =  82°  28'  34"    Check. 

A  =  45°  24'  55 


202  SPHERICAL   TRIGONOMETRY 

Ex.  2.    Given  a  =  125°  40'  14",  b  =  53°  56'  12",  c  =  98°  51'  16"; 
find  A,  B,  C. 

"~ZZ        *-*-— 

c=    98°  51'  16"  log  tan  |  =  9.65185  -  10 

2s  =  278°  27'  42"  p 

s  =  139°  13'  51"  log  tan  2  =  9.83894  -  10 


s  -  a  -  13°  33'  37"  4.  _  62o  ^  ^, 

s-b  =  So°  17'  39" 

j-c  =  40°22'85"  =24°    9'  38 


4Q/, 


log  sin  (*  -  a)  =    9.37008  -  10  C=  M 

log  sin  (s-  6)=    9.99854-10  2 

J    _  1010  QQ'   fi'f 

logsin(s-c)=    9.81145-10 

£  =  48°  19'  16" 

cologsin«=    0.12071  C  =  69°  13'  20" 

log  tan2  r  =  19.30078  -  20 
log  tan  r  =    9.65039  -  10 


EXERCISE   XXXIV 

1.  In  a  spherical  triangle  given  a=119°22'  27",  6  =  60°  44'40", 
c=108°3T'  3";  find  A,  B,  0. 

2.  In  a  spherical  triangle  given  a  =  53°  42',  b  =  118°  39'  28", 
c  =  130°  38'  20"  ;  find  A,  B,  0. 

3.  In  a  spherical  triangle  given  a  =  129°  11'  36",  b  =  109°  29' 
18",  <?=  83°  14';  find  the  largest  angle. 

4.  In  a  spherical  triangle  given  a  =  22°  56'  46",  b  =  60°  47', 
c  =  69°  49'  32"  ;  find  B  and  C. 

145.  CASE  2.  Given  two  sides  «,  6,  and  the  included  angle, 
C;  to  find  A,  B,  c.  The  angles  A,  B,  may  be  found  by  the 
first  two  of  Napier's  Analogies  (Art.  130,  p.  190)  : 

a  —  b 

A  +  B    cos—      a 

tan  =—  ot 


cos 


A-B 


SOLUTION   OF   SPHERICAL   TRIANGLES 

From  the  values  of  " — - —  and  -         -  obtained  from  these 

equations  the  values  of  A  and  B  can  at  once  be  found. 

The  value  of  c  can  then  be  obtained  from  any  one  of  Gauss's 
Equations  (Art.  129,  p.  189)  ;  for  example, 


The  solution  is  unique. 

EXERCISE  XXXV 

1.  In  a  spherical  triangle  given  a  =  85°54'  16",  6=125°  1'  27", 
<?=52°6'  26";  findJ,j&,o. 

2.  In  a  spherical  triangle  given  a  =  119°  32'  30",  6=86°31'35", 
0=49°  40'  22";  find  A,  B,  c. 

3.  In  a  spherical  triangle  given  b  =  61°  23'  18",  c  =  48°  30'  6", 
A  =  60°  53'  24"  ;  find  B,  <7,  a. 

4.  In  a  spherical  triangle  given  a  =  72°40'40",  c  =  110°33'  38", 
£=53°  50'  20";  find  -4,  0,  b. 

5.  In  a  spherical  triangle  given  b  =  68°  20'  25",  o=  52°  18'  15", 
4  =117°  12'  20";  find.£,  (7,  a. 

146.  CASE  3.  Given  two  sides  «,  6,  and  the  angle  opposite 
one  of  them  A;  to  find  B,  C  ,  c.  The  value  of  B  can  be  found 
by  means  of  the  law  of  sines  (Art.  122,  p.  182),  from  which 

we  have  •     A 

sin£  =  ^_^sin6.  (1) 

sin  a 

After  B  has  been  determined  C  and  c  can  be  found  by  the 
use  of  the  first  and  the  third  of  Napier's  Analogies. 

a  +  b 

"~ 


___tan.  (3) 

cos  - 


204  SPHERICAL   TRIGONOMETRY 

Since  £  is  determined  by  means  of  its  sine,  the  solution  is 
ambiguous. 

The  following  tests  may  be  conveniently  employed  to  deter- 
mine the  number  of  solutions. 

If  sin  A  sin  b  >  sin  a,  there  is  no  solution ;  for  in  that  case 
sin  B  >  1,  which  is  impossible. 

If  sin  A  sin  b  <  sin  #,  (1)  is  satisfied  by  two  supplementary 

values  of  B.     But  — -    -  and  -      -  are  necessarily  of  the  same 

2  2 

species.     Therefore,  if  both  these  values  of  B  satisfy  this  con- 
dition, there  are  two  solutions ;  if  not,  there  is  but  one. 

NOTE.  To  make  use  of  the  test  just  given  it  is  necessary  that  we  first 
solve  for  B.  There  are  several  methods  of  testing  for  the  number  of  solu- 
tions without  first  finding  B,  but  it  is  not  thought  best  to  include  any  of 
them  in  this  work.  For  a  full  explanation  of  them  the  student  is  referred 
to  more  extended  treatises  on  the  subject  of  Spherical  Trigonometry. 

Ex.  i.  Given  a  =  56°  30',  b  =  31°  20',  A  =  105°  11'  10"; 
find  B,  C,  c. 

Since  in  this  case  sin  A  sin  b  <  sin  a,  there  may  be  either 
one  or  two  solutions.  To  test  for  the  number  of  solutions 
we  find  the  possible  values  of  B. 

log  sin  A  =  9.98456  -  10 
log  sin  b  =  9.71602  -10 
colog  sin  a  =  0.07889 


log  sin  B  =  9.77947  -  10 

B  =  37°  0'  3", 
or,  B  =  142°  59'  57". 

We  have  from  data  given,      £±!  <  90°. 


This  shows  that  only  the  smaller  of  the  two  values  of  B  is 
admissible. 

Therefore  there  is  but  one  solution. 


SOLUTION   OF   SPHERICAL   TRIANGLES  205 

The  work  of  solution   may  be  compactly  and   conveniently 
arranged  as  follows  : 


2 

a  -  b  =  25°  10' 

^=12°  35 
A+B  =  U2°Uf  13" 

A  +  B     „, 
A-B  =  68°  II'  7"  2 


log  ain  =  9.1*501 -10 

loS  sin  =  9-34112-  10 


colog  sin  ^— ^  =  0.25140 

log  tan  ^Lzi  =  9.34874  -  10  col°S  sin  ^T  =  °'66182 

log  tan  ^9.57605 -10  log  tan  ^  =  9.83053  -  10 

log  cot  ^  =  0.33347 

-  =  20°  38'  38"  r 

±  =  24°  53'  31" 

c  =  41°  17'  16"  C  =  49°  47'  2" 

EXERCISE  XXXVI 

1.  In  a  spherical  triangle  given  a  =  71°  14',  b  =  122°  27'  18", 
'4  =  77°  23'  24";  find  B,  (7,  c. 

2.  In  a  spherical   triangle    given   a  =  80°  5'  16",   b  =  82°  4', 
.4  =  83°  34'  12";  find  B,  <7,  c. 

3.  In  a  spherical  triangle  given  a  =  151°  22'  30",  b  =  133°  31' 
25",  A=  143°  32'  28";  find  B,  0,  c. 

4.  In  a  spherical  triangle  given  a  =30°  38',  b  =  31°  29'  45", 
A  =  87°  53'  20" ;  find  the  remaining  parts. 

147    CASE  4.      Given  two  angles  A,  B,  and  the  side  oppo 
site  one  of  them  a;  to  find   C,  6,  c.     As  in  the  preceding  case 
one  of  the  parts,  in  this  case  6,  can  be  found  by  means  of  the 
law  of  sines,  from  which  we  have  (Art.  122,  p.  182) 

~~  sin  A  &1D  a ' 


206  SPHERICAL   TRIGONOMETRY 

The  values  of  c  and  O  can  then  be  found  by  means  of  the 
fourth  and  the  second  of  Napier's  Analogies  : 

.    A  +  B 

sin  - 
c 

tan2= 


sn 


C  A-B 

-*"  —  '  (3) 


The  solution  is  ambiguous,  the  value  of  b  being  determined 
by  means  of  its  sine. 

If  sin  B  sin  a  >  sin  A,  there  is  no  solution;  for  in  that  case 
sin  b  >  1,  which  is  impossible. 

If  sin  B  sin  a  <  sin  A,  (1)  is  satisfied  by  two  supplementary 
values  of  b.  To  ascertain  whether  or  not  both  these  values  are 
admissible  we  proceed  in  a  manner  similar  to  that  employed 
in  the  last  case.  If  both  values  of  b  satisfy  the  condition  im- 

posed by  the  fact  that  -  -^  —  and  a         are  of  the  same  species, 

there  are  two  solutions  ;  otherwise  there  is  but  one. 

NOTE.  The  number  of  solutions  can  always  be  determined  by  forming 
the  polar  of  the  given  triangle  and  then  determining  by  the  tests  under 
Case  3  the  number  of  solutions  of  that  triangle.  The  number  of  solutions  of 
the  given  triangle  is  always  the  same  as  the  number  of  solutions  of  its  polar. 

Ex.  i.  Given  A  =  29°  43'  12",  B  =  45°  4'  18",  a  =  36°  19'  32"; 
find  6,  c,  C. 

In  this  case  sin  B  sin  a  <  sin  A  ;  therefore  there  may  be  either 
one  or  two  solutions.  Solving  for  5,  we  proceed  as  follows  : 

log  sin  B  =  9.85003  -  10 
log  sin  a  =  9.77260  -  10 
colog  sin  A  =  0.30173 
log  sin  b  =  9.92736  -  10 

b  =  57°  48'  38", 
or,  b  =  122°  13'  22". 

We  have  from  data  given,    A  + 


SOLUTION   OF   SPHERICAL   TRIANGLES  207 

% 

Both  of  the  values  of  b  just  found  satisfy  this  condition. 
Hence,  there  are  two  solutions.  The  values  of  0  and  c  can 
now  be  found  in  the  ordinary  manner,  both  values  of  b  being 
employed. 

EXERCISE  XXXVII 

1.  In  a  spherical  triangle  given  A  =  109°  20'  10",  .#=134° 
16'  24",  a=  148°  48'  40";  find  6,  c,  0. 

2.  In  a  spherical  triangle  given  J.  =  113°  30',  .8=125°  31' 
34",  a  =  66°  44'  40";  find  5,  c,  0. 

3.  In  a  spherical  triangle  given  A  =  28°  35'  5",  J5  =  47°  51' 
15",  a  =  38°  41  '32";  find  b,  c,  0. 

4.  Iii  a  spherical  triangle  given  A  =  24°  30',  5=38°  15', 
a  =  65°  22';    find  5,  c,  0. 

148.  CASE  5.  Given  a  side  c  and  the  two  adjacent  angles 
A,  B  ;  to  find  a,  6,  C.  The  third  and  fourth  of  Napier's 
Analogies  may  be  used  for  determining  the  values  of  a  and  b 
(Art.  130,  p.  190)  : 

A-B 

cos  - 


2 
A-B 


From  these  formulas  the  values  of  a  and  b  can  be  obtained. 
The  value  of  C  can  then  be  found  by  means  of  the  first  of 
Napier's  Analogies  : 

a-b 

cos  — 

a          2      .A  +  B 

tan  -  =  --  cot 


2  a  +  b  2 

COS~2~ 
The  solution  is  unique. 


208 


SPHERICAL   TRIGONOMETRY 


Ex.  l.    Given  A  =108°  28'  55",  B  =  38°  11'  27",  c  =  52°  29'; 
find  a,  ft,  0. 


=  35°  8'  44 


=  73°  20' 11' 


r  =  26°  14'  30" 


log  cos  -  —  -  =  9.91259  -  10 


log  tan     =  9.  69282-  10 


colog  cos 


=  0.54250 


log  tan  -  —  =  10.14791  -  10 
^±-^  =  54°  34'  24.4" 
=  16°  30'  1.3" 


2 
a-b 


a  =  71°  4'  26" 
&  =  38°  4'  23" 


log  sin 


=  9.76016  -  10 


log  tan  |  =  9.69282  -  10 
colog  sin  A  +  B.  =  0.01863 


log  tan  ^~  =  9.47161  -  10 
^-=^=16°  30'  1.3" 

log  cos  —  —  =  9.98174  -  10 


lo    cot 


colog  cos 


=  9.47599  -  10 


=  0.23682 


log  tan  ^  =  9.69455  -  10 

|  =  26°  19' 56" 
C  =  52°  39'  52" 


EXERCISE  XXXVIII 

1.  In  a  spherical  triangle  given  ^.  =  126°  40'  50",  j5=81° 
45'  42",  c=  51°  56'  12";  find  a,  b,  0. 

2.  In  a  spherical  triangle  given  B=  27°  27'  36",  C  =  40°  44' 
20",  a  =155°  16';  find  6,  c,  A 

3.  In  a  spherical  triangle  given  J.  =  127°  19'  38",  (7=108° 
41'  30",  b  =  125°  22'  32";  find  a,  c,  .5. 

4.  In  a  spherical  triangle  given  A  =  154°   20'  42",  B  ==  79° 

16'  22",  c  =  85°  24'  28";  find  a,  b,  C. 

149.  CASE  6.  Given  the  three  angles  A,  B,  C;  to  find  the 
three  sides  a,  b,  c.  Any  of  the  three  groups  of  formulas  in 
Art.  128,  p.  187,  can  be  used.  The  formulas  for  the  tangents 
are  recommended  in  preference  to  those  for  the  sines  or  for  the 
cosines. 


SOLUTION   OF   SPHERICAL   TRIANGLES  209 


-       cos       - 


o 


„ 


If  all  three  of  the  sides  are  to  be  found,  it  is  convenient  to 
proceed  in  a  manner  similar  to  that  employed  in  Art.  144,  p. 
200,  where  three  sides  were  given  and  three  angles  were  to  be 
found. 

Multiplying  both  numerator  and  denominator  of  the  fraction 
under  the  radical  sign  in  (1)  by  cos  ($  —  A)  we  have 


Putting  tan  R  =  " 


^,  A)t!OS(CO8(^,  g) 

we  may  write 

tan  -  =  tan  R  cos  (S  —  A)  . 

Making  the  corresponding  changes  in  (2)  and  (3),  we  have 
the  three  equations 


tan  -  =  tan  R  cos  ($  —  J5), 
2 


tan  |  =  tan  R  cos  (#  -  (7). 

The  solution  is  unique. 
COXANT'S  TRIG.  — 14 


210  SPHERICAL   TRIGONOMETRY 

Ex.  l.     Given  A  =  221°,  B  =  128°,    0=  153°  ;  to  find  a. 
The  formula  for  tan  -,  with  the  algebraic  sign  of  each  factor  written 
above  it  for  convenience,  is  as  follows  : 


tanfl=     / 

—  cos  5  cos  (S  —  A  ) 

2     A 

_                   — 

\c 

os(S-B)cos(S- 

C) 

A 

=  221° 

log- 

cos  S  =  9.51264  - 

10 

B 

=  128° 

log  cos(S 

-.-0  =  9.93753- 

10 

C 

=  153° 

colog  cos  (S 

-  B)  =  0.26389 

2  S 

=  502°' 

colog  cos  (5 

-  C)  =  0.85644 

2)20.57050- 

20 

S 

=  251° 

s 

-A 

=  30° 

log 

tan  ^=10.28525  - 

-10 

s 

-B 

=  123° 

-  =  62°  35'  35 

o 

s 

-C 

=  98° 

a  =  125°  11'  10" 

The  result  is  real  (Art.  128,  p.  187),  the  four  negative  signs  under  the 
radical  producing  a  positive  quantity. 

Ex.  2.     Given  A  =  21°  26'  20",    B  =  56°  46'  28",    (7=115°  23' 
4";  find  a,  5,  c. 

Proceeding  by  the  second  method,  we  first  find  the  value  of  log  tan  R. 
The  following  is  suggested  as  a  convenient  arrangement  of  the  work: 


tan  R=       — 

\  cos  (S-  A)  cos  (S  -  B)  cos  (S  -  C) 

A   _  910  9f>'  ()f\r> 

log  tan  2  =  9.30865  -  10 

B  =  56°  46'  28"  2 

C  =  115°  23' 4"  log  tan*  =9.70007 -10 

25  =  193°  35'  52"  2 

5  =  96°  47'  50"  log  tan  c-  =  9.87055  -  10 
S  -  A  =  75°  21'  36" 

iS'-£  =  400    T28"  5  =  11°  30' 17.5" 
S-C  =  -19°  24'  52" 

log  cos  S  =  9.07330  -  10  o  =  :}l°  39/  43" 
colog  cos  (5  -  /I )  =  0.59732 

^_  Qf>o  o-/  4  f>// 

colog  cos  (S  -B)  =  0.11590  2  " 

colog  cos  (S  -  C)  =  0.02542  a  =  23°    0'  35" 

log  tan2  R  =  9.81 194  -  10  b  =  63°  19'  26" 

log  tan  R  =  9.90597  -  10  c  =  73°  10'    9" 


SOLUTION   OF   SPHERICAL   TRIANGLES  211 

EXERCISE   XXXIX 

1.  In  a  spherical  triangle  given  A  =  121°  40'  24",  B=  60°  12' 
22",  O=  105°  40';  find  a,  b,  c. 

2.  Iii  a  spherical  triangle  given  ,4  =  58°  20'  27",  £=84°30'30", 
(7=61°  35'  10";  find  a,  6,  c. 

3.  In  a  spherical  triangle  given  A  =  105°  14'  4",  B=55°  31' 
24",  0  =88°  51  '6";  find  a,  5,  <?. 

4.  In  a  spherical  triangle  given  A  =  171°  49'  33",  B=  5°  15'  23", 
0  =9°  18'  28";  find  a,  6,  c. 

THE   AREA   OF   A   SPHERICAL   TRIANGLE 

150.  In  considering  the  problem  of  finding  the  area  of  a 
spherical  triangle  we  have  two  principal  cases  to  consider. 

I.   Given  the  three  angles  A,  B,  C. 

Let  r  =  radius  of  sphere. 

E  =  spherical  excess  =  A  +  B  +  0  -  180°. 
A  =  area  of  triangle. 

It  is  proved  in  geometry  that  the  area  of  a  spherical  triangle 
is  to  the  area  of  the  surface  of  the  sphere  as  its  spherical  excess, 
in  degrees,  is  to  720°.  Hence,  we  have 

A  :  4  Trr2  -  E  :  720°. 


180° 

II.   Given  the  three  sides  a,  &,  c. 
The  problem  is  to  express  the  value  of  E  in  terms  of  the  sides. 

(1)    CAGNOLI'S  THEOREM. 


sin  -=  sin 


+  B  .    0  A  +  B        C 

—  sin  -  -  cos  —    —  cos  - 


212  SPHERICAL   TRIGONOMETRY 

0       C 
sin  —cos  — 

—  (cos  ^  -  cos  ^}     (Art.  130,  p.  190) 

C         \  '—  A     J 

cos- 


:    C       Ofn   -    a    .    V 
"-  —  cos-   "  "- 

2        2 


v»  \_,  /  /->  flf  •  '      0  \ 

sm— cos— (  2  sin-  sin  -  ) 


c 
cos- 


(Art. 77,  p.  100) 


. 
sin  -  sin  -  m    . 

—  _____  ^  V  sin  s  sin  ( s  —  a)  sin  (g  —  6)  sin  ( s  —  c} 
cos  c_  sin  a  sin  5 

2  (Art.  127,  p.  186) 


Replacing  sin  a  and  sin  b  by  their  values  (Art.  80,  p.  106) 
and  canceling,  we  have 

E _  Vsin  8  sin  (g  —  a)  sin  (g  —  6)  sin  (s  —  c) 

Sill  —  — .  '  —  • 

2  n         a         b         c 

L  COS  -  COS  -  COS  - 


(2>  L'HUILIER'S  THEOREM.     This  theorem,  which  expresses 
the  value  of  E  by  means  of  its  tangent,  is  derived  as  follows : 


~Ei  A 

tan  —  = — 


A  +  £  ,         TT-  (7 

cos  -       -  +  cos  - 


(Art,  77,  p.  100) 


SOLUTION   OF  SPHERICAL  TRIANGLES  213 


a-b  c          0 

cos cos  -   cos  — 

^_     2 2  (Art>  129,  p.  189) 

a  -\-b  c     •     C 

cos-^-  +  cos-   sm- 


sin sin 

__JL_— -JLcotS.  (Art- 77>  P-  10°) 


(Art.  127,  p.  186) 


(8)  All  other  cases  may  be  solved  by  first  finding  the  three 
sides  or  the  three  angles,  and  then  applying  the  proper  formula. 


ANSWERS 


1. 1. 

2.  If 

3.  0.7581+. 


PLANE   TRIGONOMETRY 
Exercise  I.     Pages  11,  12 


4.  1.2737+. 

5.  2. 54 19-. 

6.  3.5693+. 


7.    40°,  60°,  80°. 

17.  5°,  25°,  150°. 

18.  30°,  360°,  21600°. 


4. 
5. 
6. 
7. 
8. 
9. 
10. 


30°. 

120°. 

36°. 

54°. 

270°. 

150°. 

540°. 

2700°. 


11.    = 


3'   T 


Exercise  II.     Pages  14-16 

12 

2?r 

17     8599  TT 

23.    27°, 

63°. 

3 

5400 

24.    52°, 

66°, 

72°. 

3?r. 

lg     20533  TT 

7T 

7T 

7  7T 

13. 

5400 

25.    f, 

0 

3' 

Is" 

STT 

19      W7r  . 

26.    30°, 

60°, 

90°. 

14. 

• 

180 

4 

20     ^- 

27.    4,  6. 

121  7T 

180 

oa     3?r 

57T 

77T 

360 

21.    J. 

28.    --, 

7 

9 

16. 

463  TT 

oo      13021  TT 

OQ        1       T 

2?r 

1 

720 

30000 

'    2'  3 

'    3 

/      2 

5?r      2ir 

31.    150°,  —  ;  82° 

30',  11^;  135°. 

37T 

9   '     3    ' 

6 

'    24    ' 

4 

32. 


minutes  past  four  ;  54T6r  minutes  past  four. 


5.  1.77. 

6.  28°  7'  30". 

7.  0.265  sec. 

8.  40yd. 

9.  2°  8'  52.8". 

10.  861,031  mi. 
(approximately) . 

11.  3962.95. 

12.  14°  19'  26.2". 

13.  1.047  radians, 


Exercise  III.     Pages  17-19 

14.  51.56.  21.    65°  24'  30.4". 

15.  102  ft. 

(approximately) . 

16.  5:4. 


17.  3.1416. 

18.  -,  -T,  - 
399 

19.  3.1416. 

20.  0.000097+. 

215 


22.  98 ft. 

23.  1  mi.  908  ft.  nearly. 

24.  7  mi.  1237.2  ft. 

25.  18°  and  58°. 

26.  19.099'. 

27  60^ 

10800 

28.  0.00004848. 


216 


ANSWERS 


Exercise  V.     Pages  29,  30 

7.  T4rV7.  11.    Ii,  $?.  15.    U>  e 

8.  |f.  12.    |,  |.  16.    f  A/7, 

9.  f,  f.  13.    ft  A/61,  ft  A/61.  17.    f,  ft  A/14 
10.  &  A/16,  f                         14.    |Vfl,2VO. 

18.  sin  A  =  T87,  cos  A  =  tf  ,  etc.  ;  sin  5  =  jf  ,  cos  J5  =  T87,  etc. 


1».      Sill  ^1  —  

x*  +  y' 

-  ,  ws  ^i  —  •  —  ,  CIA;.  ,   »iii  jj  —  , 

V>V_/0    J-f     ~ 

20.    f. 

21.    i.                           22.    f. 

^23T}f 

Exercise  VIII.    Pages  42-48 

32.    60. 

43.    $  a2  cot  A.           53.   23°  11'  55". 

63.    355.34. 

33.   45188. 

44.    \a?\,*\\B.          54.    38°  9'  25". 

64.   74.335. 

34.    6. 

45.    £c2  sin  A  cos  A.  55.    80.49,105.64. 

65.    42.838. 

35.    124.71. 

46.    29°  22'.                56.    74°  43'  54". 

66.    313.1. 

36.    182.8. 

47.    60°  38'.                57.    124.27. 

67.    38.13. 

37.    1143.4. 

48.   20.48.                   58.    560.88. 

68.   43.03. 

38.    1916.64. 

49.   33.64.                  59.    25.165,  36.458. 

69.    39°  11'. 

39.   36157.5. 

50.   41°  36'.                60.    89.44. 

71.    118.3. 

40.   498.51. 

51.    24°  54'  16".         61.    46.71. 

72.    100. 

41.    52444.44. 

52.   42°  42'  34".        62.    122.53. 

73.    145.58. 

42.    iaVc2-a2. 

Exercise  IX.     Pages  49,  50 

1.    64°  20'  26". 
2.   75°  32'  50". 

9.    M«2sin-cos  —  . 
16. 

A  =  309.01. 
A  =  29.82. 

3.    243.57. 

10.    wrt2  sin  A  cos  A.               17. 

A  =  104.71. 

4.    175.068. 

11.    nh'2cotA.                        18. 

A  =  12.312. 

5.    148.91'. 
6.    80°  17'. 

12.    A  =  69.24.                        19. 
13.    A  =  1325.46.                    20. 

A  =  115.92. 
A  =  700.616. 

7.   91.204. 

14.    A  =  3741.18.                   21. 

A  =  2186.95. 

8.    3°  34'  8". 

Exercise  X.    Pages  72,  73 

K     V3  +  1 

„     V2-2                  Q     V3  +  2 

11.     L^. 

12.    -2. 

2 

2                                   2 

c     1  +  2  V2 

3  A/3                     1Q    3  A/3  -4. 

13.    -|. 

6            2       " 

8>       2                                        3 

,  etc. 


ANSWERS  217 

14.  Positive  for  60°,  120°,  210°,  330°  ;  negative  for  0°,  240°,  300°. 

15.  Positive  for  330°;  negative  for  210°,  300°;  zero  for  135°. 

2ab         2ab  2a  +  l  2  a2  +  2  a 


'   a2  +  62'    a2  -  62  2  a2  +  2  a  +  1 '   2  a-  +  2  a  +  1 

Exercise  XI.    Pages  83,  84 

5.  45°  and  226°  ;  45°,  135°,  22o°,  315°. 

6.  Positive  for  120°  and  690° ;  negative  for  150°,  300,  and  —  ;  zero  for  135° 
and  315°. 

7.  Positive  for  210°  and  780°;  negative  for  240°,  300°,  625°  and  — ;  zero 
for  225°. 

8.  Positive  for  60°,  150°,  and  ^^ ;  negative  for  120°  and  210° ;  zero  for  135° 

0 

and  225°. 

9.  (a)    240°  and  300°;  (6)    210°  and  330°;  (c)    135°  and  315°;  (d)   30°  and 
210°. 

14.  3. 

15.  —  cot2  A  esc  A. 

Exercise  XII.    Pages  88,  89 

•I.  14.  *  =  «»±i. 

4  6 

15.    0  =  n7r±-- 

2  ^4 

s\         /j   ( I  \  n  ^T 

±  6' 

7.  e=mr-(-\y\. 

6  17.  e  =  nr±-. 

8.  6  =  2mr±^-  18     ^  =  W7r±7r> 

7T 

9.  ^  =(2  w  +  !)TT  ± -•  v 

19.    ^  —  WTT  or  mr  ±  -  • 

2  20.    6=  mr±  -• 

4 

11.  0=(2W  +  1)7T. 

21.  <9^2w7r+-. 

12.  f  =  iw  +  7-  3 

4 

22.  »*<1.-H),  +  S. 


Exercise  XIII.     Pages  90,  91 

4.   2  nir  ±  £,  or  (2  n  +  I)TT.  5.    2  mr,  or  2  nTr  ±  |. 

3 

g      W7r  _)_  (_  1)»  ^,   or  MTT  +  (  —  1)M^-^- 
6  2 


218  ANSWERS 


7.    WT+(-l)»?. 


2  W7T 


8.  2  mr  ±  ~  18.    2  HIT,  or 

9.  2  WTT  ±  *,  or  (2  w  +  I)TT.  19.    -1^-,  Or     2  r?r 


8\  y  "  - 

m  —  n         m  +  n 

10.  2  nir  ±  -.  20.    nir  —  -,  or  —  +  — . 

3  4  3        12 

11.  2  nir  +  -,  or  sin  0  =  -  |.  21.    WTT  +  ^,  or  7-^  +  -JL. 

^j 

12.  W7T   ±   -.  22.      7i7T. 

13.  WTT  +  -,  or  cot  6  =  —  2.  23.    — . 

4  3 

14.  HT±^.  24.    5?  +  -. 


15.  (2n  +  l)or.  25.         r 

1  3  2(m  — 

16.  +,orH.  26.     (2)1  + 


Exercise  XIV.     Pages  95-97 

4-  -if-  5-   ;;:;•  6.  ^ 

Exercise  XV.     Pages  99,  100 
1.    1.  2.    &\.  3.    -  II  4.    -  4.  5.    3. 

Exercise  XVIII.     Pages  108-110 

-  •>,  u. 


,      4V2 

23 

2      7 

3\/15 

3.  it,  -  Hi 

3VI6 

9    ' 

27* 

8' 

16 

4.    |,  6£. 

• 

5 

Exercise 

XXI. 

Pages  120,  121 

1. 

±^V2, 

6       V^ 

9.    I. 

13. 

±  iv 

2. 

±  \  V2. 

2V?' 

10.    1  or  -  i. 

14. 

If- 

3. 

±1. 

7.    V3. 

11.    0  or  ±  .]. 

15. 

1- 

4. 

x  imaginary. 

-3±Vl7 

12.    1  or  \. 

16. 

*V5. 

5. 

13. 

4 

17. 

aft 

19 

ab 

20. 

V3. 

vV  — 

1  -f  Vft 

Va2- 

1  +V62-1 

21. 

2. 

18. 

WTT  or  n 

r+J, 

4 

ANSWERS  219 


Exercise  XXII.     Page  127,  128 

1.  2»r,  or2*w-  —  •  11.    mr  +(-  1)W36°  52',  or  2  mr  -£. 

3  2 

.  12.    2n7r-36°52'. 

2.  2W7T  +  -,  or  (2n 

13.  ,  or  . 
439 

14.  f,or«,±|. 

-' 


7.    2W7T  +  —  ,  or2w7r-—  . 


16. 

O 

17.    ^  +  |, 
18. 


19.    S;qr-2!:+(--I)-i. 


20.    2n7r,  or 
10.    2/i7r,  or  2  WIT  +  112°  38'. 


21.    2  mr,  (2  n  +  1)  -  ,  or  (2  n  +  1)  -• 


22.    (2w+l)^,(2rc  +  l)^,or(2N  +  l). 
'2  4  o 

23.      2  7Z7T,    01'   ?I7T  ±  y  •  o  _ 

35.    2  mr,  or  2  7i?r  +  —  • 


25.    ttr,or(2«  +  l).  ^     B)r  _  ^  or  .yr  +(_  1)njr  . 


26.  - 

27.  »,±,«r2»,±  , 


, 

4  2 

29.  n7r±^,  or(2n+l)-« 

41.  5Efor-»»±| 

30.  (2*  +  l)f,or^.f(-l)*|. 

42.  WTT,  or  nir±- 

31.  7i7T.  3 


32.    (2w+  1)|,  or  WIT  ±|- 


43.    »nr,  or 


33.    ?ITT,  or  WTT  ±     •  44.    WTT,  or     ~  + 


220 


ANSWERS 


Exercise  XXIV.     Pages  136-138 

11.  640.65ft. 

12.  AC  =  8332.2  ft.,  AB  =  12163.53  ft. 

13.  Distances  2841.2  ft.,  3475.46  ft.     Height  1721.08  ft. 

14.  Distances  11975.68  ft.,  24182.77  ft.     Height  19769.54  ft. 

15.  121.04ft.     16.    171.15ft.     17.    110.39ft.      19.    4.588  mi.      20.    4.506  mi. 


Exercise  XXVI.  Pages  146-148 

11.  4536.4  ft.  13.    5402.6  ft.  15.    15.6. 

12.  134.49  ft.      14.  9.  16.  1781.2  ft. 
19.  A=  39°  46'  0.4",  B  =  68°  2'  45.6".   20.  4494.3  ft. 


17.  5.65. 

18.  4.58:  9.81 


Exercise  XXVII.     Pages  152,  153 

17.  43°  55'  13".  20.   60°,  60°,  60°. 

18.  49°  8' 46".  21.    66°  44' 2",  60°  26' 53",  52°  49' 9". 

19.  30°,  60°,  90°.  23.    60°.          24.    120°.          25.    73°  44'. 

Miscellaneous  Examples.    Pages  158-165 

1.  247.56  ft.  3.    41°  9'  7".  5.    48°  45'  44".  7.    122.48  ft. 

2.  42°  42' 34".          4.    36°  22' 21".  6.    72.75ft.  8.    123.47ft. 
9.  Height  =  1224.3  ft.;  distance  =  1292.9  ft.                 10.    431.78  ft. 

11.  233.27  ft.  12.    440.36  Ib. ;  63°  12'  26",  26°  47'  34". 

13.  2881.46  mi.  15.    2304.52ft.  17.    7912.8  mi. 

14.  407.61ft.  16.    67.5ft.  18.    108°  11'. 
19.  Height  =  350.67  ft.,  distance  =3205.15  ft.  20.    8.0076  in. 
21.  746  ft.                               22.    17.32,  30,  34.64.              23.   244.95. 
24.  tan-1  f ;  3\  of  an  hour.                                                 25.   6ft. 

26.  136.13  ft.  from  the  foot  of  the  tower.  ,     27.   61.24ft. 

29.  109.9ft.  31.    308.66ft.  33.    110°^'.^"        35.    4782.2ft. 

30.  4621.1ft.  32.    407.61ft,  34.    473.3ft.  36.    2785.6ft. 
37.  60°  20'  8",  76°  49'  18",  42°  50'  29".                                         38.    595.84  ft. 
39.  1743.36  ft.         40.   4244.4  ft.         41.    9.1  mi.  arc-hour.        42.   383.37  yd. 

43.  Resultant  =  658.36  Ib.  ;  angle  bet.  resultant  and  greater  force  22°  23'  43". 

44.  2019.62  ft.  47.    63.08.  50.    3883  ft.  52.    13451.52  ft. 

45.  410.35ft.  48.    45.92ft.  51.    4494.3ft.  53.    1949.77ft. 

46.  178.88  ft.  49.    10520.49  ft. 


Exercise   XXVIII.     Pages  167,  168 

2.  0.0029089.          4.  0.002036.  6.  0.99999. 

3.  0.9999958.          5.  0.004363.  7.  0.00003878. 


ANSWERS  221 

Exercise  XXXI.     Page  175 

1.    r  =  2.5,  0  =  2.165.  5.    2  «c2  -  ad2  =  bed. 


2. 
3. 

4. 
9. 

10. 
11. 

a2  +  b'2  =  c2  +  cf2.                                   6.   a4  —  2  a2  +  &2  =  0. 
z2  ,  y2  _  2                                              7.    «2&  -  b  =  2. 
«2     62                                                   8.    (a2  -  &2)2  -  16  ab  =  0. 
a2  +  ^2  _  6-2  +  C2. 

(a2  +  I)2  +  2  &(a2  +  l)(a  +  6)  -  4(a  +  ft)2  =  0. 

(a  +  6)*  +  fa  _  5)1  -  2                    12-    («  ~  &)  tan  «  +  «&  =  °- 
|        i              _|                                  13.    a2  -  &2  -  2  cos  a  -  2  =  0. 

SPHERICAL   TRIGONOMETRY 

Exercise  XXXIII. 

Pages  199,  200 

1. 

6   = 

14°  25'  20", 

A  — 

47°  30'  46", 

B 

=  44°  25'  26" 

, 

2. 

c   = 

77°  56'  37", 

A  = 

81°  50'  9", 

B 

=  34°  28'  58" 

. 

3. 

Impossible. 

4. 

c  = 

69°  55'  18", 

b  = 

130°  15'  58", 

A  =  115°  33'  51". 

5. 

a  = 

49°  30'  54", 

b  = 

131°  41'  29", 

B 

=  124°  6'  53" 

, 

6. 

a  = 

34°  20'  53", 

b  = 

42°  23'  40", 

c 

=  52°  25'  39" 

7. 

a  = 

80°  28'  44", 

b  = 

78°  38'  54", 

B  =  83°  47  '23". 

8. 

b  = 

145°  13'  27" 

,  A  = 

35°  2'  7", 

B 

=  118°  8'  2". 

9. 

b  = 

155°  23'  47" 

,    c  = 

71°  18'  48", 

A 

=  98°  54'  34". 

10. 

&i  = 

153°  59'  53" 

,   Ci  = 

69°  36', 

BI 

=  152°  6'  47" 

.62  = 

26°  0'  7", 

Co   = 

110°  24',        B-2 

=  27°  53'  13" 

11. 

c  = 

62°  33'  19", 

A  = 

68°  51'  35", 

B 

=  39°  59'  48". 

12. 

a  = 

49°  53'  28", 

b  = 

58°  26', 

c 

=  70°  17'  27". 

Exercise  XXXIV. 

Page  202 

1. 

A  = 

113°  51'  22" 

,B  = 

66°  17'  20", 

C 

=  96°0'18". 

2. 

A  = 

65°  10', 

B  = 

98°  50'  37", 

C 

=  125°  17'  48". 

3. 

A  = 

129°  22'  58", 

,  B  = 

109°  41  '38", 

c 

=  97°  21'  36". 

4. 

A  = 

23°  16'  48", 

B  = 

62°  13'  34", 

G 

=  107°  54'  18' 

Exercise  XXXV. 

Page  203 

1. 

A  = 

117°  33'  50",  B  = 

46°  37'  46",  c 

— 

62°  36'  45". 

2. 

A  = 

116°  0'  7", 

B  = 

51°  34'  15",  c 

— 

57°  51'  26". 

3. 

B  = 

101°  4'  47", 

C  = 

40°  8'  22",    a 

= 

57°  31'  43". 

4. 

A  = 

35°  18'  32", 

c  = 

126°  39'  6",  b 

— 

77°  10'  36". 

5. 

B  = 

69°  28'  26", 

c  = 

42°  13'  34",  a 

= 

76°  17'  36". 

Exercise  XXXVI. 

Page  205 

1. 

B  =  119°  34'  43",    C  =  96°  55'  26", 

c  =  105°  36'  14". 

2. 

7?  =  63°1'40", 

C  =  84°  50'  28", 

c  =  80°  51'  28". 

3. 

BI  =  63°  55'  10". 

Ci  =  33°51'5",  ci  = 

=  26°  41'  4". 

B»  =  1  16°  4'  50", 
Impossible. 


222 


ANSWERS 


Exercise  XXXVII.    Page  207 

1.  6  =  156°  51'  40",  c  =  30°  57' 43", 

2.  b  =  125°  22'  40",  c  =  155°  48'  12", 

3.  bi  =  75°  38'  40",  ci  =  102°  0'  42", 

4.  b.2  =  104°  21'  20",  c2  =  134°  30'  27", 

Exercise  XXXVIII.     Page  208 

1.  a  =  129°  29'  29",  b  =  107°  45'  45", 

2.  6  =  36°  23'  38",  c  =  122°  53'  23", 

3.  «  =  123°  21'  30",  c  =  84°  15'  24", 

4.  a  =  153°  51'  21",  6  =  89°  26', 

Exercise  XXXIX.    Page  211 

1.  a  =  142°  5'  25",  b  =  38°  47'  39", 

2.  a  =  49°20'39",  6  =  62°  31' 13", 

3.  a  =  107°  45'  46",  b  =  54°  27'  19", 


4.  a  =  118°  52'  50", 


b  =  34°  20'  45", 


(7=69°  37 '20". 

C=  155°  50' 58". 
Ci  =  48°  27'  53". 
C2  =  146°  55'  13". 


C  =  54°  54' 16". 
A  =  161°  1'  28". 
5  =  129°  4' 47". 
(7=  78°  21' 23". 


c  =  135°  57'  44". 
c  =  51°37'5". 
c  =  99°  18'  46". 
c  =  84°  53'  32". 


FIVE-PLACE 

LOGARITHMIC  AND  TRIGONOMETRIC 

TABLES 

BASED   OX   THE   TABLES   OF   F.   G.    GAUSS 


ARRANGED    BY 


LEVI   L.    CONANT,  PH.D. 

PROFESSOR    OF    MATHEMATICS    IN    THE    WORCESTER 
POLYTECHNIC    INSTITUTE 


NEW    YORK  •:.  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


COPYRIGHT,  1909,  BY 

AMERICAN   BOOK   COMPANY. 

ENTERED  AT  STATIONERS'  HALL,  LONDON. 


CONANT   TRIG.    TAliLES. 

W.  P.     I 


INTRODUCTION 

1.  A  logarithm  is  the  exponent  by  which  a  number  a  must  be 
affected  in  order  that  the  result  shall  be  a  given  number  m.     That 
is,  if  ax  =  m,  then  x  is  called  the  logarithm  of  m  to  the  base  a.     The 
above  equation  written  in  logarithmic  form  is  loga  m  =  x. 

Any  positive  number  except  1  may  be  used  as  the  base  of  a 
system  of  logarithms.  In  practical  work  involving  numerical 
computation  10  is  the  base  that  is  universally  employed. 

All  computations  by  means  of  logarithms  are  based  on  the 
following  theorems  : 

2.  The  logarithm  of  a  product  is  equal  to  the  sum  of  the  logarithms 
of  the  factors. 

PROOF.  Let  m  and  n  be  any  two  positive  numbers,  and  let  x 
and  y  be  their  logarithms  respectively.  Then 

m-n  =  10*-1  O^IO^. 
/.  log(w^i)  =  x  -f-  y  =  log  m  +  log  n. 

3.  The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the 
dividend  minus  the  logarithm  of  the  divisor. 

PROOF.  ™  =  ™!  =10*-". 

n      I0y 

.'.  log  —  =  x  —  y  =  log  m  —  log  n. 
n 

4.  The  logarithm  of  any  power  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  multiplied  by  the  index  of  the  power. 

PROOF.  my  =  (10*)*  =  10**. 

.'.  log  my  =  xy  —  y  log  m. 

5.  The  logarithm  of  any  root  of  a  number  is  equal  to  the  logarithm 
of  the  number  divided  by  the  index  of  the  root. 

PROOF.  Vm  =  ^10^  =  10*. 

]ow  m 


6.  The  logarithm  of  any  integral  power  or  root  of  10  is  an 
integral  number.     The  logarithms  of  all  other  positive  numbers 
are  fractions. 

Negative  numbers  have  no  logarithms.  If  any  logarithmic 
computation  is  to  be  performed  which  involves  negative  numbers, 
the  problem  should  be  solved  as  though  the  numbers  were  all  posi- 
tive ;  and  the  algebraic  sign  of  the  result  should  then  be  deter- 
mined by  the  usual  methods  of  algebra. 

7.  The  logarithm  of  a  number  consists  of  two  parts,  an  integral 
part  and  a  decimal.     The  integral  part  is  called  the  characteristic, 
and  the  decimal  part  the  mantissa.     As  logarithms  are   usually 
printed  the  mantissa  is  always  positive.     The  characteristic  may 
be  positive,  negative,  or  zero.     The  characteristic  of  the  logarithm 
of  any  number  may  be  found  by  one  of  the  following  rules  : 

I.  The  characteristic  of  the  logarithm  of  a  number  greater  than 
one  is  positive,  and  is  one  less  than  the  number  of  digits  in  the  integral 
part  of  the  number. 

II.  The  characteristic  of  the  logarithm  of  a  decimal  fraction  is 
negative,  and  is  numerically  one  greater  than  the  number  of  ciphers 
immediately  after  the  decimal  point. 

For  example,  the  characteristic  of  the  logarithm  of  3286  is  3  : 
of  294645  is  5  ;  of  0.0241  is  -2  ;  of  0.000649  is  -4. 

For  the  sake  of  convenience  a  negative  characteristic  is  often 
changed  in  form  by  adding  to  it  and  subtracting  from  it  the  number 
10.  For  example,  if  the  characteristic  of  a  logarithm  is  —  2,  and 
the  mantissa  is  .38416,  the  logarithm  may  be  written  8.38416  —  10. 
If  the  characteristic  is  —  1  and  the  mantissa  is  .74925,  the  logarithm 
may  be  written  9.74925  —  10.  If  the  negative  forms  of  the  charac- 
teristics are  retained,  the  above  logarithms  are  written  2.38416  and 
1.74925  respectively.  When  it  is  remembered  that  the  mantissas 
are  positive,  the  reason  for  writing  the  negative  sign  of  a  charac- 
teristic above  instead  of  before  it  will  be  readily  understood. 

In  all  work  connected  with  the  logarithms  in  the  following 
tables  the  characteristics,  when  negative,  are  to  be  understood  as 
being  increased  and  diminished  by  10. 

TABLE    I 

Directions  for  finding  the  logarithm  of  a  number. 

8.  When  the  number  is  between  i  and  100. 

The  entire  logarithm,  including  both  characteristic  and  man- 
tissa, is  given  on  p.  9. 


9.  Numbers  containing  one  or  two  significant  figures. 

The  mantissa  is  found  on  p.  9.  It  is  the  same  for  all  numbers 
containing  the  same  significant  figures  arranged  in  the  same  order, 
no  matter  where  the  decimal  point  is  placed. 

The  characteristic  is  found  by  means  of  the  rules  given  above. 

For  example, 

log  53  =  1.72428,  log  .53  =  9.72428  -  10, 

log  5.3  =  0.72428,  log  .053  =  8.72428  -  10. 

10.  Numbers  containing  three  significant  figures. 

The  number,  no  attention  being  paid  to  the  decimal  point,  is 
found  at  the  left  of  the  page  in  the  column  headed  No.  The 
mantissa  is  found  on  a  line  with  the  number,  and  in  the  column 
headed  0.  The  characteristic  is  found  as  before,  by  one  or  the 
other  of  the  rules  on  p.  4. 

For  example, 

log  763  =  2.88252,  log  .0763  =  8.88252  -  10, 

log  76.3  =  1.88252,  log  .00763  =  7.88252  -  10. 

11.  Numbers  containing  four  significant  figures. 

The  first  three  significant  figures  are  found  in  the  column 
headed  No.,  and  the  fourth  is  at  the  top  of  the  page.  On  a 
line  with  the  first  three  figures,  and  in  the  column  headed  by  the 
fourth  figure,  the  mantissa  is  found.  The  characteristic  is  deter- 
mined as  in  the  previous  cases. 

For  example, 

log  296300  =  5.47173,         log  .2963  =  9.47173  -  10, 
log    29,63=1.47173,    log  .0002963  =  6.47173  -  10. 

12.  Numbers  containing  more  than  four  significant  figures. 

Let  the  number  whose  logarithm  is  required  be  61487.  Since 
the  number  lies  between  61480  and  61490,  the  logarithm  of  the  re- 
quired number  lies  between  the  logarithms  of  those  numbers,  i.e. 
between  4.78873  and  4.78880. 

Now  log  61490  =  4.78880 

and  log  61480  =  4.78873 

giving  a  difference  of  .00007 

Hence,  we  see  that  an  increase  of  10  in  the  number  produces 
an  increase  of  .00007  in  the  logarithm.  But  the  actual  increase 
we  have  to  consider  in  the  number  is  7.  Now  if  an  increase  of 
10  in  the  number  produces  an  increase  of  .00007  in  the  logarithm, 


an  increase  of  7  in  the  number  will  produce  an  increase  of  -^  of 
.00007,  or  .000049.     Calling  this  correction  .00005,  we  have 

log  61480  =  4. 78873 

correction  =    .00005 

.-.  log  61487  =  4.78878 

It  is  here  assumed  that  an  increase  in  the  number  is  accom- 
panied by  a  proportional  increase  in  the  logarithm  of  the  number. 
This  is  not  true  ;  but  in  obtaining  logarithms  from  a  table,  that 
assumption  is  always  made.  If  greater  accuracy  is  desired,  it 
will  be  necessary  to  use  tables  containing  a  greater  number  of 
figures. 

Directions  for  finding  the  number  corresponding  to  a  given 
logarithm. 

13.  Logarithms  whose  mantissas  are  found  in  the  table. 
When  the  exact  mantissa  of  a  logarithm  is  found  in  the  table, 

the  first  three  significant  figures  of  the  number  corresponding  to 
the  logarithm  are  found  in  the  column  headed  No.,  and   on  a 
line  with  the  given  mantissa.     The  fourth  significant  figure  is  at 
the  top  of  the  column  in  which  the  given  mantissa  is  found. 
For  example, 

2.68529  is  the  logarithm  of  484.5.     See  p.  17. 
9.68529-10  is  the  logarithm  of  0.4845. 
7.68529  -  10  is  the  logarithm  of  0.004845. 
5.68529  is  the  logarithm  of  484500. 

14.  Logarithms  whose  mantissas  are  not  found  in  the  table. 

When  the  exact  mantissa  of  the  given  logarithm  is  not  found 
in  the  table,  the  first  four  significant  figures  of  the  number  corre- 
sponding to  the  logarithm  are  the  same  as  the  first  four  significant 
figures  of  the  number  corresponding  to  the  next  smaller  logarithm. 
The  remaining  figures  are  found  by  interpolation,  as  illustrated  in 
the  following. 

To  find  the  number  corresponding  to  the  logarithm  3.44127. 

Number  corresponding  to  3.44138  is  2763     See  p.  13. 
Number  corresponding  to  3.44122  is  2762 

.00016          ~T 

Thus  we  see  that  an  increase  of  .00016  in  the  logarithm  corresponds 
to  an  increase  of  1  in  the  number.  But  the  given  logarithm, 
3.44127,  is  .00005  greater  than  the  logarithm  of  the  number  2762. 


Therefore,  the  increase  in  the  required  number  is  ;$$$}f,  or,  more 
simply,  -^g-  of  1.  This  gives  .31  as  the  required  increase.  Hence 
2762.31  is  the  number  whose  logarithm  is  3.44127. 

Similarly, 

78.565  is  the  number  whose  logarithm  is  1.89523. 

58317.5  is  the  number  whose  logarithm  is  4.76580. 

.17532  is  the  number  whose  logarithm  is  9.24383  -  10. 

15.    Cologarithms. 

The  cologarithm  of  a  number  is  the  logarithm  of  the  recipro- 
cal of  that  number. 

Since  the  reciprocal  of  a  number  is  unity  divided  by  that  num- 
ber, and  since  the  logarithm  of  unity  is  0,  it  follows  that  the 
cologarithm  of  a  number  is  found  by  subtracting  the  logarithm  of 
the  number  from  0,  or  from  10  —  10. 

For  example, 
colog  256  =  log  2  iff  =  log  1  -  log  256  =  0  -  2.40824  =  -  2.40824. 

To  avoid  the  use  of  negative  logarithms  the  above  work  is 
performed,  and  the  value  of  the  above  result  is  expressed  as 

follows:  log  1  =  10. 00000 -10 

log  256=    2.40824 
.-.  colog  256=    7.59176-10. 

From  the  definition  of  a  cologarithm  it  follows  that  the  effect 
of  subtracting  the  logarithm  of  a  number  is  the  same  as  that  of 
adding  its  cologarithm.  For  example,  finding  the  logarithm 
of  HI  by  each  of  the  two  possible  methods,  we  have  : 

BY  LOGARITHMS  BY  COLOGAKITHMS 

log  293  =  12.46687  -  10  log  293  =  2.46687 

log  478=    2.67943  colog  478=  7.32057  -  10 

Subtracting,      9.78744  -  10  Adding,     9.78744  -  10 
The  result  is  the  same  in  both  cases. 

TABLE   III 

This  table  contains  the  logarithmic  sine  and  tangent  for  every 
second  from  0'  to  3',  and  the  logarithmic  cosine  and  cotangent  for 
every  second  from  89°  57'  to  90°  ;  the  logarithmic  sine,  cosine,  and 
tangent  for  every  ten  seconds  from  0°  to  2°,  and  the  logarithmic 
sine,  cosine,  and  cotangent  for  every  ten  seconds  from  88°  to  90° ; 
and  the  logarithmic  sine,  cosine,  tangent,  and  cotangent  for  every 
minute  from  1°  to  89°. 
I 


16.  The   logarithmic   sine,  cosine,  tangent,  or  cotangent  of  an 
angle  less  than  90°. 

If  the  angle  is  less  than  45°,  use  the  column  having  the  name 
of  the  proper  function  at  the  top,  and  the  column  of  minutes  at 
the  left  of  the  page;  if  the  angle  is  between  45°  and  90°,  use  the 
column  having  the  name  of  the  proper  function  at  the  bottom, 
and  the  column  of  minutes  at  the  right  of  the  page. 

To  illustrate  the  use  of  this  table,  let  us  find  the  logarithm  of 
sin  26°  28'  12". 

%  P-  48>  log  sin  26°  28'  =  9.64902  -  10. 

The  difference  between  log  sin  26°  28'  and  log  sin  26°  29'  is  .00025. 
Increasing  the  former  by  ^|  of  this  difference,  or  .00005,  we  have 

log  sin  26°  28'  12"  =  9.64907  -  10. 
As  a  further  illustration,  find  log  tan  71°  3£/  10". 

%  P-  44>         log  tan  71°  38'  =  10.47885  -  10. 
Increasing  this  by  J  J  of  .00042,  i.e.  by  .00013,  we  have 
log  tan  71°  38'  19"  =  10.47898  -  10. 

If  the  logarithm  of  a  cosine  or  of  a  cotangent  is  to  be  found, 
the  correction  for  seconds  must  be  subtracted,  since  these  functions 
decrease  as  the  angle  increases  from  0°  to  90°. 

17.  The  angle  corresponding  to  a  logarithmic  sine,  cosine,  tan- 
gent, or  cotangent. 

Find  the  angle  whose  log  tan  =  9.65647  —  10. 

The  next  smaller  logarithmic  tangent  is  (p.  47)  9.65636  —  10, 
which  corresponds  to  an  angle  of  24°  23'.  The  difference  between 
this  logarithm  and  the  log  tan  24°  2i'  is  .00033,  and  the  difference 
between  log  tan  24°  23'  and  the  given  logarithm  is  .00011.  There- 
fore, the  angle  corresponding  to  the  next  smaller  logarithm,  i.e. 
24°  23',  must  be  increased  by  1J  of  60",  i.e.  by  20".  Hence, 
9.65647  -  10  =  log  tan  24°  23'  20". 

In  the  case  of  the  logarithm  of  the  cosine  or  of  the  cotangent 
we  work  from  the  next  larger  logarithm  to  the  next  smaller,  in- 
stead of  from  the  smaller  to  the  larger  as  in  the  case  of  the  sine 
and  the  tangent. 

TABLE    IV 

This  table  contains  the  numerical  or  natural  values  of  the  sine, 
cosine,  tangent,  and  cotangent  for  every  minute  from  0°  to  90°. 


—                                                                       - 

TABLE  I 

THE   COMMON    OR    BRIGGS 

LOGARITHMS 

OF   THE    NATURAL    NUMBERS 

FEOM  1   TO   10000 

MOO 

No.        Log. 

No.        Log. 

No.        Log. 

No.         Log. 

No.        Log. 

2O     1.30103 
21     1.32222 
22     1.34242 
23     1.36173 
24     1.38021 

4O  1.60206 
41  1.61278 
42  1.62325 
43  1.  63  347 
44  1.64345 

6O     1.77815 

61     1.78533 
62     1  .  79  239 
63     1.  79  934 
64     1.  80  618 

8O     1.90309 
81     1.90849 
82     1.91381 
S3     1.  91  908 
84     1.92428 

1     0.  00  000 
2    0.  30  103 
3     0.47712 
4    0.60206 

5     0.  69  897 
6    0.77815 
7    0.84510 
8    0.90309 
9    0.  95  424 

25     1.  39  794 
26     1.41497 
27     1.43136 
28     1.44716 
29     1.46240 

45  1.  65  321 
46  1.66276 
47  1.67210 
48  1.68124 
49  1.69020 

65     1.  81  291 
66     1.  81  954 
67     1.82607 
68     1.83251 
69     1.  83  885 

85     1.  92  942 
86     1.93450 
87"  1.93952 
88     1.94448 
89     1.94939 

MBh   i.ooooo 

11     1.  04  139 
12     1.07918 
13     1.  11  394 
14     1.  14  613 

3O     1.47712 
31     1.  49  136 

32     1.50515 
33     1,51851 
34     1.53148 

5O  1.69897 
51  1.70757 
52  1.71600 
53  1.  72  428 
54  1.  73  239  * 

7O     1.84510 
71     1.85126 
72     1.85733 
73     1.86332 
74     1.86923 

9O     1.95424 
91     1.95904 
92     1.  96  379 

93     1.968-JS 
94     1.97313 

15     1.  17  609 
16     1.20412 
17     1.  23  045 
18     1.2-5527 
19     1.27875 

35     1.54407 
36     1.  55  630 
37     1.  56  820 
38     1.57978 
39     1.  59  106 

55  1.  74  036 
56  1.74819 
57  1.75587 
58  1.76343 
59  1.77085 

75     1.  87  506 
76     1.  88  081 
77     1.88649 
78     1.89209 
79     1.  89  763 

95     1.  97  772 
96     1.  98  227 
97     1.98677 
98     1  .  99  123 
99     1.  99  564 

2O     1.  30  103 

4O     1.60206 

6O     1.77815 

8O     1.90309 

1OO    2.00000 

MOO 


It) 


100-149 


No. 

01234 

56789 

1OO 

00000   00043    00087   00130   00173 

00217    00260   00303    00346   00389 

101 

00432    00475    00518   00561    00604 

00647    00689   00732   00775    00817 

102 

00860   00903    00945    00988   01030 

01072    01115    01157    01199   01242 

103 

01  284    01  326    01  368    01  410   01  452 

01494    01536    01578    01620   01662 

104 

01  703    01  745    01  787    01  828   01  870 

01912   01953    01995    02036   02078 

105 

02  119   02  160   02  202    02  243    02  284 

02325    02366   02407   02449   02490 

106 

02531    02572    02612    02  653    02  694 

02735    02776   02816   02857    02898 

107 

02938   02979   03019   03060   03100 

03141    03181    03222   03262   03302 

108 

03342   03383    03423    03463    03503 

03  543    03  583    03  623    03  663    03  703 

109 

03743    03782   03*822   03862    03902 

03941    03981-04021    04060   04100 

110 

04139   04179   0-1218   04258  04297 

04336   04376   04415    04454   04493 

111 

04532    04571    04610   04650   04689 

04727   04766   04805    04844   04883 

112 

04922    04961    04999   05038    05077 

05  115    05  154   05  192   05  231    05  269 

113 

05  308   05  346   05  385    05  423    05  461 

05500   05538   05576   05614   05652 

114 

05  690   05  729   05  767   05  805    05  843 

05881    05918    05956   05994   06032' 

\ 

115 

06070   06108   06145    06183    06221 

06258   06296   06333    06371    06408 

116 

06446    06483    06521    06558    06595 

06633    06670   06707   06744   06781 

117 

06819   06856   06  893    06930   06967 

07004    07-041    07078   07115    07151 

118 

07  188   07  225    07  262    07  298    07  335 

07372   07408   07445    07482   07518 

119 

07555    07591    07628   07664   07700 

07737   07773    07809   07846   07882 

12O 

07918   07954   07990   08027   08063 

08099   08135    08171    08207    08243 

121 

08279   08314   OS  3"50   08386   08422 

08458   08493    08529   08565    08600 

122 

08636   08672   08707    08743    08778 

08814    08849   08884    08920   08955 

123 

08991    09026    09061    09096   09132 

09167.09202    09237    09272    09307 

124 

09342   09377   09412   09447   09482 

09517   09552   09587   09621    09656 

125 

09691    09726   09760   09795    09830 

09864   09899   09934   09968    10003 

126 

10037    10072    10106    10140    10175 

10209    10243    10278    10312    10346 

127 

10380    ]0415    10449    10483    10517 

10551    10585    10619    10653    10687 

128 

10721    10755    10789    10823    10857 

10890    10924    10958    10992    11025 

129 

11  059    11  093    11  126    11  160    11  193 

11227    11261    11294    11327    11361 

13O 

11394    11428    11461    11494    11528 

11561    11594    11628    11661    11694 

131 

11727    11760    11793    11826    11860 

11893    11926    11959    11992    12024 

132 

12057    12090    12123    12156    12189 

12222    12254    12287    12320    12352 

133 

12385    12418    12450    12483    12516 

12548    12581    12613    12646    12678 

134 

12710    12743    12775    12808    12840 

12872    12905    12937    12969    13001 

135 

13033    13066    13098    13130    13]62 

13194    13226    13258    13290    13322 

136 

13354    13386    13418    13450    13481 

13513    13545    13577    13609    13640 

137 

13672    13704    13735    13767    13799 

13830    13862    13893    13925    13956 

138 

13988    14019    14051    14082    14114 

14145    14176    14208    14239    14270 

139 

14301    14333    14364    14395    14426 

14457    14489    14520    14551    I45^H 

140 

14613    14644    14675    14706   14737 

14768    14799    14829    14860    1489^ 

141 

14922    14953    14983    15014    15045 

15  076    15  106    15  137    15  168    15  198 

142 

15229    15259    15290    15320    15351 

15381    15412    15442    15473    15503 

143 

15534    15564    15594    15625    15655 

15685    15715    15746  15  776    15806 

144 

15836   15866    15897    159^7    15957 

15987    16017    16047    16077    16107 

145 

16137    16167    16197    16227    16256 

16286   16316   16346   16376   16406 

-  146 

16435    16465    16495    16524    16554 

16584    16613    16643    16673    16702 

147 

16732    16761    16*791    16820    16850 

16879    16909    16938    16967    16997 

148 

17026    17056    17085    17114    17143 

17173    17202    17231    17260    17289 

149 

17319    17348    17377    17406    17435 

17464    17493    17522    17551    17580 

No. 

O          1*2          3          4 

56789 



100-149 


150-199 


ii 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150 

17609 

17638 

17667 

17696 

17725 

17754 

17  782 

17811 

17840 

17869 

151 

17898 

17926 

17  955 

17984 

18  013 

18041 

18070 

18099 

18127 

18156 

152 

18184 

18213 

18241. 

18270 

18298 

18327 

18355 

18384 

18412 

18441 

153 

18469 

18498 

18526 

18554 

18  583 

18611 

18639 

18667 

18  696 

18724 

^154 

18752 

18780 

18808 

18837 

18865 

18893 

18921 

18949 

18977 

19005 

155 

19033 

19061 

19089 

19117 

19145 

19173 

19201 

19229 

19257 

19285 

156 

19312 

19340 

19368 

19  396 

19424 

19451 

19479 

19507 

19535 

19562 

157 

19590 

19618 

19645 

19673 

19700 

19728 

19756 

19783 

19811 

19838 

158 

19866 

19893 

19921 

19  948 

19976 

20003 

20030 

20  058 

20085 

20112 

159 

20140 

20167 

20194 

20222 

20249 

20276 

20303 

20330 

20358 

20385 

16O 

20412 

20439 

20466 

20493 

20520 

20548 

20575 

20602 

20629 

20  656 

161 

20683 

20710 

20737 

20763 

20790 

20817 

20844 

20871 

20898 

20925 

162 

20  952 

20978 

21005 

21  032 

21059 

21085 

21  112 

21139 

21  165 

21  192 

163 

21219 

21  245 

21272 

21299 

21325 

21  352 

21378 

21405 

21  431 

21458 

164 

21484 

21511 

21537 

21  564 

21590 

21617 

21643 

21669 

21696 

21722 

165 

21  748 

21775 

21801 

21  827 

21  854 

21880 

21906 

21932 

21  958 

21985 

166 

22011 

22  037 

22063 

22  089 

22115 

22  HI 

22167 

22194 

22220 

22246 

167 

22272 

22298 

22324 

22  350 

22376 

22  401 

22427 

22453 

22479 

2T2505 

168 

22531 

22557 

22  583 

22608 

22634 

22660 

22686 

22712 

22737 

22763 

169 

22789 

22814 

22840 

22866 

22891 

22917 

22943 

22968 

22994 

23019 

17O  23045 

23070 

23096 

23121 

23  If  7 

23172 

23198 

23223 

23249 

23274 

171 

23300 

23325 

23  350 

23376 

23401 

23426 

23  452 

23477 

23,502 

23528 

172 

23553 

23378 

23603 

23629 

23  654 

23679 

23  704 

23729 

23754 

23779 

173 

23805 

23830 

23855 

23  880 

23905 

23930 

23955 

23980 

24005 

24030 

174 

24055 

24080 

24105 

24130 

24155 

24180 

24204 

24229 

24254 

24279 

175 

24304 

24  329 

24353 

24378 

24403 

24428 

24452 

24477 

24  502 

24527 

176 

24551 

24576 

24601 

24625 

24650 

24674 

24699 

24724 

24748 

24773 

177 

24797 

24822 

24846 

24871 

24895 

24920 

24944 

24969 

24993 

25018 

178 

25  042 

25066 

25091 

25115 

25  139 

25  164 

25188 

25  212 

25  237 

25261 

179 

25285 

25310 

25  334 

25358 

25  382 

25406 

25431 

25455 

25479 

25503 

180 

25527 

25  551 

25575 

25600 

25624 

25  648 

25672 

25696 

25720 

25744 

181 

25768 

25  792 

25816 

25  840 

25864 

25888 

25  912 

25  935 

25  959 

25983 

182 

26  007 

26031 

26055 

26079 

26102 

26126 

26150 

26174 

26198 

26221 

183 

26245 

26269 

.26293 

26316 

26340 

26364 

26387 

26411 

26435 

26458 

184 

26482 

26  505 

26529 

26  553 

26576 

26600 

26623 

26647 

26670 

2§  694 

185 

26717 

26741 

26764 

26788 

26811 

26834 

26858 

26881 

26905 

26928 

1B6 

26951 

26975 

26998 

27021 

27045 

27068 

27091 

27114 

27138 

27161 

^gjl 

^27184 

27207 

27231 

27254 

27277 

27300 

27323 

27346 

27370 

27393 

M:  416 

27439 

27462 

27485 

27508 

27531 

27554 

27577 

27600 

27623 

i    V;  646 

27669 

27  692 

27715 

27738 

27761 

27784 

27807 

27830 

27852 

W^  27  875 

27898 

27921 

27944 

27967 

27989 

28012 

28035 

28058 

28081 

191  1  28  103 

28126 

28149 

28171 

28194 

28217 

28240 

28262 

28285 

28307 

192 

28330 

28  353 

28375 

28398 

28421 

28443 

28466 

28488 

28511 

28533 

193 

28556 

28  578 

28601 

28623 

28646 

28668 

28691 

28713 

28735 

28758 

194 

28780 

28803 

28825 

28847 

28870 

28892 

28914 

28937 

28959 

28981 

195 

29003 

29026 

29048 

29070 

29092 

29115 

29137 

29  159 

29181 

29203 

196 

29226 

29248 

29270 

29292 

29314 

29  336 

29  358 

29380 

29403 

29425 

197 

29  447 

29469 

29491 

29  S13 

29535 

29  557 

29579 

29601 

29623 

29645 

198 

29667 

29688 

29710 

29732 

29754 

29776 

29798 

29820 

29842 

29  863 

199 

29885 

29907 

29929 

29951 

29973 

29994 

_3a?i?^ 

30016 

30038 

30060 

30081 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150-199 


12 


200-249 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2OO 

30103 

30125 

30146 

30168 

30190 

30211 

30233 

30255 

30276 

30298 

201 

30320 

30341 

30363 

30384 

30406 

30428 

30449 

30471 

30492 

30514 

202 

30  535 

30557 

30578 

30  600 

30621 

30643 

30664 

30685 

30707 

30728 

203 

30750 

30771 

30792 

30814 

30835 

30  856 

30878 

30899 

30920 

30942 

204 

30963 

30984 

31006 

31027 

31048 

31069 

31091 

31112 

31  133 

31154 

205 

31  175 

31  197 

31218 

31239 

31260 

31281 

31302 

31323 

31345 

31366 

206 

31387 

31408 

31  429 

31450 

31471 

31492 

31  513 

31534 

31  555 

31576 

207 

31  597 

31618 

31  639 

31660 

31  681 

31  702 

31  723 

31  744 

31765 

31  785 

208 

31806 

31  827 

318-18 

31869 

31  890 

31911 

31931 

31  952 

31973 

31994 

209 

32015 

32035 

32056 

32077 

32098 

32118 

32139 

32160 

32181 

32201 

21O 

32222 

32  243 

32263 

32284 

32305 

32  325 

32346 

32  366 

32387 

32408 

211 

32428 

32449 

32469 

32490 

32510 

32  531 

32  552 

32572 

32593 

32613 

212 

32634 

32654 

32  675 

32  695 

32715 

32736 

32  756 

32777 

32797 

32818 

213 

32838 

32  858 

32879 

32  899 

32  919 

32940 

32  960 

32980 

33001 

33021 

214 

33041 

33062 

33082 

33102 

33122 

33143 

33  163 

33183 

33203 

33224 

215 

33244 

33264 

33284 

33304 

33325 

33  345 

33  365 

33385 

33405 

33425 

216 

33445 

33465 

33  486 

33  506 

33  526 

33  546 

33  566 

33  5S6V 

33  606 

33  626 

217 

33646 

33666 

33  686 

33  706 

33  726 

33  746 

33766 

33786 

33  806 

33826 

218 

33  846 

33866 

33885 

33  905 

33  925 

33  945 

33  965 

33985 

34005 

34025 

219 

34044 

34064 

34  084 

34104 

34124 

34143 

34163 

34183 

34  203 

34223 

22O 

34242 

34  262 

34282 

34301 

34  321 

34  341 

34361 

34380 

34400 

34  420 

221 

344^9 

34  459 

34479 

34498 

345ft 

34  537 

34  557 

34  577 

34  596 

34  616 

222 

34635 

34  655 

34674 

34  694 

34713 

34  733 

34753 

347^2 

34  792 

34811 

223 

34830 

34850 

34869 

34889 

34  908 

34  928 

34  947 

34  967 

34  986 

35  005 

224 

35025 

35044 

35  064 

35083 

35102 

35  122 

35  141 

35160 

35  ISO 

35  199 

225 

35  218 

35  238 

35  257 

35  276 

35295 

35315 

35334 

35353 

35  372 

35  392 

226 

35411 

35430 

35  449 

35  468 

35  488 

35  507 

35  526 

35  545 

35564 

35583 

227 

35603 

35  622 

35  641 

35  660 

35  679 

35  698 

35717 

35736 

35  755 

35  774 

228 

35  793 

35813 

35  832 

35851 

35  870 

35  889 

35908 

35  927 

35  946 

35965 

229 

35  984 

36003 

36021 

36040 

36059 

36078 

36  097 

36116 

36135 

36  154 

23O 

36  173 

36192 

36211 

36229 

36248 

36  267 

36  £86 

36305 

36324 

36342 

231 

36361 

36380 

36399 

36418 

36  436 

36  455 

36474 

36  493 

36511 

36  530 

232 

36  549 

36  568 

36  586 

36605 

36  624 

36  642 

36  661 

36  680 

36698 

36  717 

233 

36736 

36  754 

36  773 

36  791 

36810 

36  829 

36847 

36  866 

36  884 

36  903 

234 

36922 

36940 

36959 

36^977 

36  996 

37014 

37033 

37051 

37070 

37088 

235 

37107 

37  125 

37  144 

37162 

37  181 

37  199 

37218 

37236 

37254 

37  273 

236 

37  291 

37310 

37328 

37346 

37365   37383 

37401 

37420 

37  438 

37457 

237 

37475 

37493 

37511 

37  530 

37  548   37  566 

37  585 

37  603 

37621 

238 

37  658 

37  676 

37  694 

37712 

37  731 

37  749 

37  767 

37  785 

378031 

239 

37840 

37858 

37876 

37  894 

37912 

37931 

37949 

37967 

37  98| 

24O 

38021 

38  039 

38057 

38  075 

38093 

38112 

38130 

38148 

38166 

241 

38202 

38220 

38  238 

38  256 

38274 

38  292 

38310 

38328 

38  346 

38  364 

242 

38382 

38399 

38417 

38435 

38  453 

38471 

38  489 

38  507 

38  525 

38  543 

243 

38  561 

38578 

38  596 

38614 

38  632 

38  650 

38668 

38686 

38703 

38721 

244 

38739 

38  757 

38775 

38792 

38810 

38828 

38846 

38863 

38881 

38899 

245 

38917 

33  934 

38952 

38970 

38987 

39005 

39023 

39041 

39  058 

39076 

246 

39  094 

39111 

39129 

39  146 

39164 

39182 

39199 

39217 

39  235 

39  252 

247 

39^70 

39287 

39305 

39  3-22 

39  340 

39  358 

39375 

39  393 

39410 

39428 

248 

39  445 

39  463 

39480 

39498 

39515 

39  533 

39  550 

39  568 

39  585 

39602 

249 

39620 

39637 

39655 

39672 

39690 

39707 

39724 

39742 

39  759 

39  777 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200-249 


250-299 


13 


No. 

O           1           2           3          4 

56789 

25O 

39794   39811    39829   39846   39863 

39881    39898   39915    39933    39950 

251 

39967    39985    40002    40019   40037 

40054    40071    40088    40106    40123 

252 

40140    40157    40175    40192   40209 

40226   40243    40261    40278    40295 

253 

40312    40329   40346   40364    40381 

40398    40415    40432    40449    40466 

N254 

40483    40500   40518   40535    40552 

40  569   40  586,  40  603    40  620    40  637 

255 

40654    40671    40688   40705    40722 

40739   40756   40773    40790    40807 

256 

40824    408-11    40858    40875    40892 

40909   40926   40943    .40960    40976 

257 

40993    41010    41027    41044    41061 

41078    41095    41111    47128    41145 

258 

41162    41179    41196   41212    41229 

41  246   41  263    41  280    41  296    41  313 

259 

41330   41347   41  363    41  380   41397 

41414   41430   41447    41464    41481 

26O 

41  497   41  514   41  531    41  547   41  564 

41581    41597   41614   41631    41647 

261 

41664    41681    41697    41714   41731 

41747   41764   41780   41797    41814 

262 

41  830   41  847    41  863    41  880   41  896 

41  913    41  929    41  946    41  963    41  979 

263 

41996   42012    42029   42045    42062 

42078   42095    42111    42127    42144 

264 

42  160   42  177    42  193    42  210    42  226 

42  243    42  259   42  275    42  292    42  308 

265 

42725    42341    42357    42374   42390 

42406    42423    42439    42455    42472 

266 

42488    42504    42521    42537    42553 

42570    42586   42602    42619    42635 

267 

42651    42667    42684    42700   42716 

42732    42749   42765    42781    42797 

268 

42813    42830   42846    42862    42878 

42894   42911    42927    42943    42959 

269      42975    42991    43008    43024    43040 

43  056   43  072   43  088    43  104    43  120 

27O 

43  136   43  152    43  169   43  185    43  201 

43217   43233    43249   43265    43281 

271 

43297    43313    43329   43345    43361 

43377    43393    43409    43425    43441 

272 

43457    43473    43489   43505    43521 

43537    43553    43569    43584    43600 

273 

43  616   43  632    43  648    43  664    43  680       43  696   43  712    43  727    43  743'   43  759 

274 

43775    43791    43807    43823    43838 

43854   43870   43886    43902    43917 

275 

43933    43949   43965    43981    43996 

44012   44028    44044    44059   44075 

276 

44  091    44  107    44  122    44  138   44  154 

44170   44185    44201    44217    44232 

277 

44248    44264    44279   44295    44311 

44326   44342    44358   44'373    44389 

278 

44404    44420   44436   44451    44467 

44483    44498    44514    44529    44545 

279 

44560   44576   44592   44607    44623 

44638   44654   44669    44685    44700 

28O 

44  716   44  731    44  747    44  762    44  778 

44793   44809   44824    44840   44855 

281 

44871    44886   44902    44917    44932 

44948    44963    44979    44994    45010 

282 

45025    45040   45056   45071    45086 

45102    45117    45133    45148    45163 

283  !    45  179    45  194    45  209    45  225    45  240 

45255    45271    45286    45301    45317 

284!    45332    45347   45362    45378   45393 

45408    45423    45439    45454    45469 

285  1    45  484   45500   45515    45530    45545 

45561    45576   45591    45606   45621 

286 

45637    45652    45667    45682    45697 

45  712    45*28    45  743    45  758    45  773 

287 

45  788    45  803    45  818    45  834   45  849 

45864    45879    45894    45909    45924 

288 

45  939   45  954    45  969    45  984    46  000 

46015    46030    46045    46060    46075 

289 

46090   46105    46120   46135    46150 

46165    46180    46195    46210   46225 

29O 

46240   46255    46270   46285    46300 

46315    46330    46345    46359    46374 

291 

46389    46404    46419   46434    46449 

46464    46479    46494    46509    46523 

292 

46538    46553    46568    46583    46598 

46613    46627    46642    46657    46672 

293 

46687    46702    46716    46731    46746 

46761    46776    46790    46805    46820 

294 

46  835    46  850   46  864    46  879   46  894 

46909   46923    46938    46953    46967 

295 
296 

46982    46997    47012    47026   47041 
47129   47144    47159    47173    47188 

47056   47070   47lfc   47100    47  !M 
47202    47217    47232    47246    47  2B1 

297 

47276    47290   47305    47319    47334 

47349    47363    47378    47392    47407 

298 

47422    47436   47451    47465    47480 

47494   47509   47524    47538    47553 

299 

47567    47582    47596    47611    47625 

47640   47654    47669    47683    47698 

No. 

01234 

56789 

250-299 


14 


300-349 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

300 

47712 

47727 

47741 

47756 

47770 

47784 

47799 

47813 

47828 

47  842 

301 

47  857 

47871 

47885 

47900 

47914 

47929 

47  94.3 

47958 

47972 

47986 

302 

48001 

48  015 

48029 

48  044 

48058 

48073 

48087 

48101 

48116 

48130 

303 

48144 

48159 

48173 

48187 

48202 

48216 

48230 

48244 

48  259 

48273 

304 

48287 

48302 

48316 

48330 

48344 

48359 

48373 

48387 

48401 

48416 

305 

48430 

48444 

48458 

48473 

48487 

48501 

48515 

48530 

48544 

48  558 

306 

48572 

48586 

48601 

48615 

48629 

48643 

48657 

48671 

48686 

48700 

307 

48714 

48728 

48742 

48756 

48770 

48785 

48799 

48813 

48  827 

48  841 

308 

48  855 

48869 

48  883 

48897 

48911 

48926 

48  940 

48  954 

48968 

48982 

309 

48996 

49010 

49024 

49038 

49052 

49066 

49080 

49094 

49108 

49122 

31O 

49136 

49150 

49164 

49178 

49192 

49206 

49220 

49234 

49248 

49262 

311 

49  276 

49290 

49304 

49318 

49332 

49346 

49360 

49374 

49388 

49402 

312 

49  415 

49429 

49443 

49457 

49471 

49485 

49499 

49513 

49527 

49  541 

313 

49  554 

49  568 

49582 

49  596 

49610 

49624 

49638 

49651 

49665 

49679 

314 

49693 

49707 

49721 

49734 

49748 

49762 

49776 

49790 

49803 

49817 

315 

49831 

49845 

49859 

49872 

49886 

49900 

49914 

49927 

49941 

49955 

316 

49969 

49982 

49996 

50010 

50024 

50037 

50051 

50  065 

50079 

50092 

317 

50  106 

50  120 

50133 

50  147 

50161 

50174 

50188 

50  202 

50  215 

50  229 

318 

50  243 

50  256 

50270 

50284 

50297 

50311 

50325 

50  338 

50352 

50365 

319 

50379 

50393 

50406 

50420 

50433 

50447 

50461 

50474 

50488 

50501 

320 

50515 

50529 

50542 

50  556 

50  569 

50583 

50  596 

50610 

50623 

50637 

321 

50651 

50664 

50678 

50691 

50705 

50718 

50  732 

50745 

50759 

50772 

322 

50  786 

50799 

50813 

50  826 

50  840 

50853 

50  866 

50880 

50893 

50907 

323 

50  920 

50934 

50947 

50961 

50974 

50987 

51001 

51014 

51028 

51041 

324 

51  055 

51068 

51081 

51095 

51108 

51121 

51135 

51148 

51162 

51175 

325 

51188 

51202 

51  215 

51228 

51242 

51  255 

51268 

51282 

51295 

51308 

326 

51322 

-51335 

51348 

51  362 

51  375 

51388 

51402 

51415 

51428 

51441 

327 

^1415/51468 

51481 

51495 

51508 

51521 

51534 

51548 

51561 

51574 

328 

sStffefc 

51601 

51614 

51627 

51  640 

51654 

51667 

51680 

51693 

51706 

329 

l'5l'£20 

51733 

51746 

51759 

51772 

51786 

51799 

51812 

51825 

51838 

33O 

51  851 

51865 

51  878 

51891 

51904 

51917 

51930 

51943 

51957 

51970 

331 

51983 

51996 

52009 

52022 

52  035 

52  048 

52061 

52075 

52088 

52101 

332 

52114 

52127 

52140 

52  153 

52  166 

52179 

52192 

52205 

52218 

52231 

333 

52244 

52257 

52  270 

52284 

52  297 

52310 

52323 

52336 

52  349 

52362 

334 

52375 

52388 

52401 

52414 

52427 

52  440 

52453 

52466 

52479 

52492 

335 

52504 

52  517 

52530 

52543 

52556 

52569 

52582 

52595 

52608 

52621 

336 

52634 

52647 

52  $60 

52673 

52686 

52699 

52711 

52724 

52737 

52  750 

337 

52763 

52776 

52789 

52802 

52815 

52827 

52840 

52853 

52866 

52879 

338 

52892 

52905 

52917 

52  930 

52943 

52  956 

52  969 

52982 

52994 

53007 

339 

53020 

53033 

53046 

53058 

53071 

53084 

53097 

53]  10 

53122 

53135 

340 

53148 

53161 

53  173 

53186 

53199 

53  212 

53224 

53237 

53250 

53263 

341 

53275 

53  288 

53  301 

53314 

53326 

53  339 

53352 

53364 

53377 

53390 

342 

53403 

53415 

53428 

53441 

53  453 

53466 

53479 

53491 

53504 

53  517 

343 

53529 

53542 

53555 

53567 

53580 

53  593 

53  605 

53  618 

53631 

53  643 

344 

53656 

53668 

53681 

53694 

53706 

53719 

53732 

53744 

53757 

53769 

345 

53782 

53794 

53807 

53820 

53832 

53845 

53  857 

53870 

53882 

53895 

346 

53908 

53920 

53  933 

53945 

53958 

53  970 

53  983 

53  995 

54008 

54020 

347 

54033 

54045 

54058 

54070 

54083 

54095 

54108 

54120 

54133 

54  145 

348 

54  158 

54170 

54183 

54  195 

54208 

54220 

54  233 

54  245 

54258 

54270 

349 

54283 

54295 

54307 

54320 

54332 

54345 

54357 

54370 

54382 

54394 

No. 

0 

1 

2 

3 

4 

g 

6 

7 

8 

9 

300-349 


350-399 


15 


No. 

O          1           2          3          4 

56789 

350 

351 
352 
353 
354 

54407  54419  54  432  \5  4  444  54456 
54531  54543  54555  54568  54580 
54654  54667  54679  54691  54704 
54  777  54  790  54  802  54  814  54  827 
54900  54913  54925  54937  54949 

54469   54481    54494    54506    54518 
54593    54605    54617    54630   54642 
54716    54728    54741    54753    54765 
54839    54851    54864    54876    54888 
54962    54974    54986   54998    55011 

355 
356 

357 
358 
359 

55023  55035  55047  55060  55072 
55  145  55  157  55  169  55  182  55  194 
55267  55279  55291  55303  55315 
55388  55400  55413  55425  55437 
55509  55522  55534  55546  55558 

55084    55096    55108    55121    55133 
55206    55218    55230   55242    55255 
55328    55340    55352    55364    55376 
55449    55461    55473    55485    55497 
55570    55582    55594    55606    55618 

36O 

361 
362 
363 
364 

55630  55642  55654  55666  55678 
55751  55763  55  775  55  787  55799 
55871  55883  55895  55907  55919 
55991  56003  56015  56027  56038 
56110  56122  56134  56146  56158 

55  691    55  703    55  715    55  727    55  739 
55811    55823    55835    55*847    55859 
55931    55943    55955    55967    55979 
56050    56062    56074    56086   56098 
56170    56182    56194    56205    56217 

365 
366. 
367 
368 
369 

56229  56241  56253  56265  56277 
56348  56360  56372  56384  56396 
56467  56478  56490  56502  56514 
56585  56597  56608  56620  56632 
56703  56714  56726  56738  56750 

56289    56301    56312    56324    56336 
56407    56419    56431    56443    56455 
56526    56538    56549    56561    56573 
56644    56656    56667    56679   56691 
56761    56773    56785    56797    56808 

37O 

371 
372 
373 
374 

56820  56832  56844  56855  56867 
56937  56949  56961  56972  56984 
57054  57066  57078  57089  57101 
57171  57183  57194  57206  57217 
57287  57299  57310  57322  57334 

56879    56891    56902    56914    56926 
56996    57008    57019   57031    57043 
57113    57124    57136   57148    57159 
57229    57241    57252    57264    57276 
57345    57357    57368    57380   57392 

375 
376 
377 
378 
379 

57403  57415  57426  57438  57449 
57519  57530  57542  57553  57565 
57634  57646  57657  57669  57680 
57749  57761  57772  57784  57795 
57864  57875  57887  -57898  57910 

57461    57473    57484   57496   57507 
57576    57588    57600    57611    57623 
57692    57703    57715    57726   57738 
57807    57818    57830   57841    57852 
57921    57933    57944    57955    57967 

38O 

381 

'382 
383 
384 

57978  57990  58001  58013  58024 
58092  58104  58115  58127  58138 
58206  58218  58229  58240  58252 
58320  58331  58343  58354  58365 
58433  58444  58456  58467  58478 

58035    58047    58058   58070   58081 
58149    58161    58172    58184    58195 
58263    58274    58286   58297    58309 
58377    58388    58399    58410    58422 
58490    58501    58512    58524    58535' 

385 
386 
387 
388 
389 

58546  58557  58569  58580  58591 
58659  58670  58681  58692  58704 
58771  58782  58794  58805  58816 
58883  58894  58906  58917  58928 
58995  59006  59017  59028  59040 

58602    58614    58625    58636    58647 
58715    5^726    58737    58749   58760 
58827    58838    58850    58861    58872 
58939    58950    58961    58973    58984 
59051    59062    59073    59084    59095 

39O 

391 
392 
393 
394 

59106  59118  59129  59140  59151 
59218  59229  59240  59251  59262 
59329  59340  59351  59362  59373 
59439  59450  59461  59472  59483 
59550  59561  59572  59583  59594 

59162    59173    59184^59195    59207 
59273    59284    59295    59306    59318 
59384    59395    59406    59417    59428 
59494   59506    59517    59528   59539 
59605    59616    59627    59638    59649 

395 
396 
397 
398 
399 

59660  59671  59682  59693  59704 
59770  59780  59791  59802  59813 
59879  59890  59901  59912  59923 
59988  59999  60010  60021  60032 
60097  60108  60119  60130  60141 

59715   59726   59737   59  74S-.597Sa 
59824    59835    59846    59857    5956$ 
59934    59945    59956   59966    59977 
60043    60054   60065    60076   60016 
60152    60163    60173    60184    60195 

No. 

O           1           2           3          4 

56789 

350-399 


lli 


400-449 


]$0. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

4OO 

60206 

60217 

60228 

60239 

60249 

60260 

60271 

60282 

60293 

60304 

401 

60314 

60325 

60  336 

60347 

60358 

60369 

60379 

60390 

60401 

60412 

402 

60423 

60  433 

60  444 

60455 

60466 

60477 

60487 

60498 

60509 

60520 

403 

60531 

60541 

60552 

60  563 

60574 

60584 

60595 

60606 

60617 

60627 

404 

60638 

60649 

60660 

60670 

60681 

60692 

60703 

60713 

60724 

60735 

405 

60746 

60  756 

60767 

60778 

60788 

60799 

60810 

60821 

60831 

60842 

406 

60  853 

60863 

60874 

60  885 

60895 

60906 

60917 

60927 

60938 

60949 

407 

60959 

60970 

60981 

6Q991 

61002 

61013 

61023 

61034 

61045 

61  055 

408 

4  61  066 

61077 

61087 

61  098 

61  109 

61  119 

61  130 

61140 

61  151 

61  162 

409 

61  172 

61183 

61194 

61204 

61215 

61225 

61236 

61247 

61257 

61268 

41O 

61  278 

61289 

61300 

61310 

61321 

61331 

61342 

61  352 

61363 

61374 

411 

61  384 

61  395 

61  405 

61416 

61426 

61437 

61448 

61458 

61469 

61479 

412 

61490 

61500 

61511 

61521 

61532 

61542 

61553 

61563 

61  574 

61584 

413 

61  595 

61606 

61616 

61627 

61637 

61648 

61  658 

61  669 

61679 

61690 

414 

61700 

61  711 

61  721 

61731 

61742 

61752 

61763 

61773 

61784 

61  794 

415 

61805 

61815 

61826 

61836 

61  847 

61  857 

61  868 

61878 

61888 

61899 

416 

61909 

61920 

61930 

61941 

61951 

61962 

61972 

61982 

61  993 

62003 

^417 

62014 

62024 

62034 

62045 

62  055 

62066 

62  076 

62086 

62097 

62107 

418 

62118 

62128 

62138 

62149 

62  159 

62170 

62180 

62190 

62201 

62211 

419 

62221 

62232 

62242 

62252 

62263 

62273 

62284 

62294 

62304 

62315 

420 

62325 

62335 

62346 

62356 

62366 

62377 

62387 

62397 

62408 

62418 

421 

62428 

62439 

62  449 

62  459 

62469 

62  480 

62490 

62500 

62511 

62521 

422 

62  531 

62  542 

62552 

62562 

62  572 

62  583 

62593 

62603 

62613 

62  624 

423 

62  634 

62644 

62655 

62665 

62675 

62  685 

62696 

62  706 

62716 

62726 

424 

62  737 

62747 

62757 

62  767 

62778 

62  788 

62  798 

62808 

62818 

62829 

425 

62  839 

62849 

62859 

62870 

62880 

62890 

62900 

62910 

62921 

62931 

-126 

62941 

62951 

62961 

62972 

62  982 

62992 

63  002 

63012 

63022 

63  033 

427 

63043 

63053 

63  063 

63  073 

63  083 

63094 

63  104 

63114 

63  124 

63134 

428   63  141 

63  155 

63165 

63175 

63185 

63  195 

63  205 

63215 

63  225 

63236 

429   63  246 

63256 

63266 

63276 

63286 

63296 

63306 

63317 

63327 

63337 

430 

63347 

63357 

63367 

63377 

63  387 

63397 

63407 

63417 

63428 

63438 

431 

63  448 

63458 

63468 

63478 

63  488 

63  498 

63  508 

63  518 

63  528 

63538 

432 

63548 

63  658 

63568 

63579 

63  589 

63  599 

63  609 

63619 

63629 

63639 

433 

63649 

63  659 

63669 

63679 

63  689 

63  699 

63709 

63719 

63729 

63739 

434 

63749 

63759 

63769 

63779 

63789 

63  799 

63809 

63819 

63829 

63  839 

435 

63  849 

63  859 

63869 

63  879 

63889 

63899 

63909 

63919 

63929 

63939 

436 

63949 

63  959 

63  969 

63979 

63  988 

63  998 

64008 

64018 

64028 

64  038 

437 

64048 

64058 

64068 

64  078 

64088 

64098 

64108 

64118 

64128 

64137 

438 

64  147 

64  157 

64167 

64  177- 

64  187 

64  197 

64207 

64217 

64227 

64237 

439 

64  246 

64256 

64266 

64276 

64286 

64296 

64306 

64316 

64326 

64335 

44O 

64  345 

64  355 

64365 

64375 

64385 

64395 

64  404 

64414 

64424 

64434 

441 

64  444 

64  454 

64464 

64  473 

64483 

64493 

64  503 

64513 

64  523 

64532 

442 

64542 

64  552 

64562 

64572 

64582 

64  591 

64601 

64611 

64621 

64631 

443 

64640 

64650 

64660 

64670 

64680 

64689 

64699 

64709 

64719 

64729 

444 

64738 

64748 

64758 

64768 

64777 

64787 

64797 

64807 

64816 

64826 

445 

64836 

64846 

64856 

64  865 

64  875 

64885 

64895 

64904 

64914 

64  924 

446 

64933 

64943 

64953 

64963 

64972 

64  982 

64992 

65  002 

65011 

65021 

447 

65031 

65  040 

65050 

65  060 

65  070 

65  079 

65  089 

65  099 

65108 

65  118 

448 

65128 

65  137 

65147 

65  157 

65  167 

65  176 

65  186 

65  196 

65  205 

65  215 

449 

65  225 

65  234 

65244 

65254 

65  263 

65273 

65283 

65292 

65302 

65312 

1 
No.    O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

I 

400-449 


450-499 


17 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

65  321 

65  331 

65  341 

65350 

65360 

65369 

65  379 

65  389 

65  398 

65  408 

451 

65418 

65  427 

65  437 

65  447 

65  456 

65466 

65  475 

65485 

65495 

65  504 

452 

65  514 

65  523 

65  533 

65  543 

65  552 

65  562 

65571 

65581 

65  591 

65600 

453 

65  6  10 

65  619 

65  629 

65  639 

65648 

65  658 

65667 

65677 

65686 

65696 

454 

65  706 

65715 

65725 

65734 

65744 

65753 

65763 

65772 

65782 

65792 

\ 

455 

65  801 

65811 

65  820 

65  830 

65839 

65  849 

65  858 

65868 

65877 

65887 

456 

65  896 

65  906 

65916 

65  925 

65  935 

65944 

65  954 

65963 

65  973 

65982 

457 

65  992 

66001 

66011 

66020 

66  030 

66039 

66049 

66  058 

66068 

66  077 

458 

66087 

66096 

66  106 

66115 

66124 

66134 

66143 

66153 

66162 

66172 

459 

66181 

66191 

66200 

66  210 

66219 

66229 

66238 

66247 

66257 

66266 

46O 

66  276 

66285 

66295 

66304 

66314 

66323 

66332 

66342 

66351 

66361 

461 

66  370 

66380 

66389 

66398 

66408 

66417 

66427 

66436 

66445 

66  455 

462 

66  464 

66  474 

66483 

66492 

66502 

66511 

66521 

66  530 

66539 

66549 

463 

66  558 

66  567  " 

66577 

66  586 

66  596 

66605 

66614 

66624 

66633 

66642 

464 

66652 

66  661 

66671 

66680 

66689 

66  699 

66  708 

66717 

66727 

66736 

465 

66  745 

66  755 

66764 

66773 

66783 

66792 

66801 

66811 

66820 

66829 

466 

66839 

66848 

66857 

66867 

66876 

66  885 

66894 

66904 

66913 

66922 

467 

66932 

66  941 

66950 

66960 

66969 

66978 

66987 

66997 

67006 

67015 

468 

67025 

67034 

67043 

67  052 

67062 

67071 

67080 

67089 

67099 

67108 

469 

67  1.17 

67127 

67136 

67145 

67154 

67164 

67173 

67182 

67191 

67201 

470 

67210 

67219 

67228 

67237 

67247 

67  256 

67265 

67274 

67284 

67293 

471 

67  302 

67311 

67321 

67330 

67339 

67348 

67357 

67367 

67376 

67385 

472 

67394 

67403 

67413 

67422 

67431 

67  440 

67449 

67  459 

67468 

67477 

473 

67486 

67495 

67  504 

67514 

67523 

67  532 

67541 

67  550 

67  560 

67569 

474 

67578 

67587 

67596 

67605 

67614 

67624 

67633 

67642 

67651 

67660 

475 

67669 

67679 

67688 

67697 

67706 

67715 

67724 

67733 

67742 

67  752 

476 

67761 

67770 

67  779 

67788 

67797 

67806 

67815 

67825 

67834 

67843 

477 

67852 

67861 

67870 

67879 

67888 

67897 

679J&6 

67  916 

67925 

67934 

478 

67  943 

67  952 

67961 

67  970 

67979 

67988 

67997 

68006 

68015 

68024 

479 

68  034 

68043 

68052 

68061 

68070 

68079 

68088 

68097 

68106 

68115 

480 

68124 

68133 

68  142 

68  151 

68160 

68169 

68178 

68187 

68196 

68  205 

481 

68215 

68224 

68  233 

68242 

68  251 

68260 

68269 

68278 

68287 

68296 

482 

68  305 

68314 

68323 

68332 

68341 

68350 

68  359 

68368 

68377 

68386 

483 

68395 

68404 

68  413 

68422 

68431 

68440 

68449 

68  458 

68467 

68476 

484 

68485 

68494 

68502 

68511 

68520 

68529 

68538 

68547 

68556 

68565 

485 

68574 

68  583 

68  592 

68601 

68610 

68619 

68628 

68637 

68646 

68655 

486 

68664 

68673 

68681 

68690 

68  699 

68708 

68717 

68726 

68735 

68744 

487 

68753 

68762 

687T1 

68780 

68789 

68797 

68806 

68815 

68824 

68833 

488 

68  842 

68851 

68860 

68869 

68878 

68.886 

68  895 

68904 

68913 

68922 

489 

68931 

68940 

68949 

68  958 

68  966 

68  975 

68984 

68993 

69002 

69011 

49O 

69020 

69028 

69037 

69046 

69055 

69064 

69073 

69082 

69090 

69  099 

491 

69  108 

69117 

69126 

69135 

69  144 

69  152 

69161 

69170 

69179 

69188 

492 

69197 

69  205 

69214 

69223 

69232 

69  241 

69249 

69  258 

69267 

69276 

493 

69  285 

69294 

69  302 

69311 

69320 

69329 

69338 

69346 

69  355 

69364 

494 

69373 

69381 

69390 

69  399 

69408 

69417 

69425 

69434 

69443 

69452 

495 

69461 

69469 

69478 

69487 

69496 

69  504 

69513 

69522 

69531 

69  539 

496 

69  548 

69  557 

69  566 

69574 

69  583 

69  592 

69601 

69  609 

69618 

69627 

497 

69636 

69644 

69  653 

69662 

69671 

69679 

69688 

69697 

69  705 

69  714 

498 

69723 

69732 

69740 

69749 

69758 

69  767 

69775 

69  784 

69793 

69801 

499 

69810 

69819 

69  827 

69836 

69845 

69  854 

69862 

69871 

69880 

69888 

No. 

O 

1 

2 

3 

4 

5 

O 

7 

—  8 

9 

450-499 


18 


500-549 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5OO 

69  897 

69906 

69914 

69923 

69932 

69940 

69949 

69958 

69966 

69  975 

501 

69984 

69992 

70001 

70-010 

70018 

70027 

70036 

70044 

70053 

70062 

502 

70070 

70079 

70088 

70096 

70105 

70114 

70122 

70131 

70140 

70148 

503 

70  157 

70165 

70174 

70183 

70191 

70200 

70209 

70217 

70  226 

70234 

504 

70243 

70252 

70260 

70269 

70278 

70286 

70295 

70303 

70312 

70321 

505 

70329 

70338 

70346 

70355 

70364 

70372 

70381 

70389 

70398 

70406 

506 

70415 

70424 

70432 

70441 

70449 

70  458 

70467 

70475 

70484 

70492 

507 

70  501 

70509 

70518 

70526 

70535 

70544 

70  552 

70561 

70569 

70578 

508 

70586 

70  595 

70603 

70612 

70621 

70629 

70  638 

70  646 

70655 

70663 

509 

70672 

70680 

70689 

70697 

70706 

70714 

70723 

70731 

70740 

70749 

510 

70757 

70766 

70774 

70783 

70791 

70800 

70808 

70817 

70825 

70834 

511 

70842 

70851 

70859 

70868 

70  876 

70885 

70893 

70  902 

70910 

70919 

512 

70927 

70935 

70944 

70  952 

70961 

70969 

70  978 

70986 

70995 

71003 

513 

71012 

71020 

71029 

71037 

71046 

71054 

71  063 

71071 

71079 

71088 

514 

71096 

71105 

71113 

71122 

71130 

71  139 

71  147 

71155 

71164 

71172 

515 

71181 

71189 

71198 

71206 

71214 

71223 

71231 

71240 

71248 

71  257 

516 

71265 

71273 

71282 

71290 

71  299 

71  307 

71315 

71324 

71332 

71341 

517 

71  349 

71357 

71366 

71374 

71383 

71391 

71399 

71408 

71416 

71  425 

518 

71433 

71  441' 

71450 

71458 

71466 

71475 

71483 

71492 

71  500 

71508 

519 

71517 

71525 

71533 

71  542 

71  550 

71559 

71567 

71575 

71584 

71592 

520 

71600 

71609 

71617 

71625 

71634 

71642 

71650 

71659 

71667 

71675 

521 

71684 

71692 

71700 

71709 

71  717 

71725 

71734 

71742 

71750 

71759 

522 

71767 

71775 

71784 

71792 

71800 

71809 

71817 

71825 

71834 

71842 

523 

71850 

71858 

71867 

71875 

71  883 

71892 

71900 

71908 

71917 

71925 

524 

71933 

71941 

71950 

71958 

71966 

7197,5 

71983 

71991 

71999 

72008 

525 

72016 

72024 

72032 

72041 

72049 

72057 

72066 

72074 

72082 

72090 

526 

72099 

72107 

72115 

72123 

72132 

72140 

72148 

72156 

72165 

72173 

527 

72181 

72189 

72198 

72206 

72214 

72222 

72  230 

72239 

72247 

72255 

528 

72263 

72272 

72280 

72288 

72296 

72304 

72313 

72321 

72329 

72337 

529 

72346 

72354 

72362 

72370 

72378 

72357 

72  395 

72403 

72411 

72419 

53O 

72428 

72436 

72444 

72452 

72460 

72469 

72477 

72485 

72493 

72501 

531 

72  509 

72518 

72  526 

72534 

72  542 

72  550 

72  558 

72567 

72575 

72583 

532 

72591 

72599 

72607 

72616 

72624 

72  632 

72  640 

72648 

72656 

72665 

533 

72673 

72681 

72689 

72697 

72705 

72713 

72722 

72  730 

72738 

72746 

534 

72754 

72762 

72770 

72779 

72787 

72795 

72803 

72811 

72819 

72827 

535 

72  835 

72843 

72852 

72860 

72868 

72876 

72884 

72892 

72900 

72908  1 

536 

72916 

72925 

72933 

72941 

72949 

72957 

72965 

72973 

72981 

72989 

537 

72997 

73006 

73014 

73022 

73030 

73038 

73046 

73054 

73062 

73070 

538 

73078 

73086 

73094 

73  102 

73  111 

73119 

73127 

73135 

73143 

73151 

539 

73159 

73167 

73175 

73183 

73191 

73199 

73207 

73215 

73223 

73231 

54O 

73239 

73247 

73  255 

73263 

73272 

73280 

73288 

73296 

73304 

73312 

541 

73320 

73328 

73336 

73344 

73  352 

73360 

73368 

73376 

73384 

73392 

542 

73  400 

73408 

73416 

73424 

73432 

73440 

73448 

73456 

73464 

73472 

543 

73480 

73488 

73496 

73  504 

73512 

73  520 

73528 

73536 

73  544 

73552 

544 

73560 

73568 

73576 

73  584 

73592 

73600 

73608 

73616 

73624 

73632 

545 

73  640 

73648 

73  656 

73664 

73  672 

73679 

73687 

73695 

73703 

73711 

546 

73719 

73727 

73  735 

73743 

73751 

73759 

73767 

73775 

73783 

73791 

547 

73  799 

73807 

73815 

73823 

73830 

73838 

73846 

73  854 

73862 

73870 

548 

73878 

73  886 

73894 

73902 

7391$ 

73918 

73926 

73933 

73941 

73  949 

549 

73957 

73  965 

73973 

73981 

73989 

73997 

74005 

74013 

74020 

74028 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

500-549 


550-599 


19 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

550 

74036 

74044 

74052 

74060 

74068 

74076 

74084 

74092 

74099 

74107 

551 

74115 

74123 

74131 

74139 

74147 

74155 

74162 

74170 

74178 

74186 

552 

74194 

74202 

74210 

74218 

74225 

74233 

74241 

74249 

74257 

74265 

553 

74273 

74280 

74  288 

74296 

74304 

74312 

74320 

74327 

74335 

74  343 

554 

74351 

74359 

74367 

74374 

74382 

74390 

74398 

74406 

74  414' 

74421 

\ 

555 

74429 

74437 

74445 

74453 

74461 

74468 

74476 

74484 

74492 

74500 

556 

74507 

74515 

74523 

74531 

74539 

74547 

74  554 

74562 

74570 

74578 

557 

74  586 

74593 

74601 

74609 

74617 

74624 

74632 

74640 

74648 

74656 

558 

74663 

74671 

74679 

74  687 

74695 

74702 

74710 

74718 

74726 

74733 

559 

74741 

74749 

74757 

74764 

74772 

74780 

74788 

74796 

74803 

74811 

560 

74819 

74827 

74834 

74842 

74850 

74  858 

74865 

74873 

74881 

74889 

561 

74896 

74904 

74912 

74920 

74927 

74935 

74943 

74950 

74  958 

74966 

562 

74974 

74981 

74  989 

74997 

75005 

75  012 

75020 

75028 

75035 

75  043 

563 

75  051 

75  059 

75  066 

75074 

75082 

75089 

75097 

75105 

75  113 

75120 

564 

75128 

75  136 

75  143 

75  151 

75159 

75166 

75174 

75  182 

75  189 

75197 

565 

75205 

75213 

75220 

75  228 

75236 

75243 

75251 

75259 

75266 

75274 

566 

75282 

75289 

75297 

75305 

75312 

75320 

75328 

75335 

75343 

75351 

567 

75358 

75366 

75374 

75  381 

75389 

75397 

75404 

75412 

75420 

75427 

568 

75435 

75  442 

75  450 

75  458 

75465 

75473 

75481 

75488 

75496 

75504 

569 

75511 

75519 

75526 

75534 

75542 

75549 

75  557 

75565 

75572 

75580 

570 

75  587 

75595 

75603 

75610 

75618 

75626 

75633 

75641 

75648 

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571 

75  664 

75  671 

75679 

75686 

75694 

75702 

75709 

75717 

75  724 

75732 

572 

75740 

75747 

75  755 

75762 

75  770 

75  778 

75  785 

75793 

75800 

75808 

573 

75815 

75823 

75831 

75838 

75846 

75853 

75861 

75868 

75876 

75884 

574 

75891 

75899 

75906 

75914 

75921 

75929 

75937 

75944 

75952 

75959 

575 

75967 

75974 

75982 

75989 

75997 

76005 

76012 

76020 

76027 

76035 

576 

76042 

76050 

76057 

76065 

76072 

76080 

76087 

76095 

76103 

76110 

577 

76118 

76125 

76133 

76140 

76148 

76155 

76163 

76170 

76178 

76185 

578 

76193 

76200 

76208 

76215 

76223 

76230 

76238 

76245 

76  253 

76260 

579 

76268 

76275 

76283 

76290 

76298 

76305 

76313 

76320 

76328 

76  335 

580 

76343 

76  350 

76358 

76365 

76373 

76380 

76388 

76395 

76403 

76410 

581 

76418 

76  425 

76433 

76440 

76448 

76455 

76462 

76470 

76477' 

76485 

582 

76492 

76500 

76  507 

76515 

76522 

76530 

76537 

76  545 

76552 

76559 

583 

76567 

76574 

76  582 

76  589 

76597 

76604 

76612 

76619 

76626 

76634 

584 

76641 

76649 

76  656 

76664 

76671 

76678 

76686 

76693 

76701 

76708 

585 

76716 

76723 

76730 

76738 

76  745 

76753 

76760 

76768 

76775 

76782 

586 

76790 

76797 

76805 

76812 

76819 

76827 

76834 

76842 

76849 

76856 

587 

76864 

76871 

76  879 

76886 

76  893 

76901 

76908 

76916 

76923 

76930 

588 

76938 

76945 

76  953 

76  960 

76967  ' 

76975 

76982 

76989 

76997 

77004 

589 

77012 

77019 

77026 

77034 

77041 

77048 

77056 

77063 

77070 

77078 

59O 

77085 

77093 

77100 

77107 

77115 

77122 

77129 

77137 

77144 

77151 

591 

77  159 

77166 

77173 

77181 

77188 

77195 

77203 

77210 

77217 

77225 

592 

77232 

77240 

77247 

77  254 

77262 

77269 

77276 

77283 

77291 

77298 

593 

77  305 

77313 

77320 

77327 

77335 

77342 

77349 

77357 

77  364 

77371 

594 

77379 

77386 

77393 

77401 

77408 

77415 

77422 

77430 

77437 

77444 

595 

77  452 

77  459 

77  466 

77  474 

77481 

77488 

77495 

77503 

77510 

77517 

596 

77525 

77532 

77539 

77  546 

77  554 

77  561 

77  568 

77576 

77583 

77590 

597 

77597 

77605 

77612 

77619 

77627 

77634 

77641 

77648 

77  656 

77663 

598 

77670 

77677 

77685 

77692 

77699 

77706 

77714 

77721 

77728 

77  735 

599 

77743 

77750 

77757 

77764 

77772 

77779 

77786 

77793 

77801 

77808 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

550-599 


20 


600-649 


No. 

O 

1 

2 

3 

4 

5 

G 

7 

8 

9 

GOO 

77815 

77822 

77830 

77837 

77  844 

77  851 

77  859 

77866 

77873 

77  880 

601 

77887 

77895 

77902 

77909 

77916 

77924 

77931 

77938 

77  945 

77  952 

602 

77960 

77  967 

77974 

77981 

77  988 

77996 

78003 

78010 

78017 

78  025 

603 

78032 

78  039 

78046 

78053 

78  061 

78068 

78075 

78082 

78089 

78  097 

604 

78104 

78111 

78118 

78125 

78132 

78140 

78147 

78154 

78161 

78168 

605 

78176 

78183 

78190 

78197 

78204 

78211 

78219 

78226 

78233 

78240 

606 

78247 

78254 

78262 

78269 

78276 

78283 

78290 

78297 

78305 

78312 

607 

78319 

78326 

78333 

78340 

78347 

78355 

78362 

78369 

78376 

78  383 

608 

78390 

78398 

78405 

78412 

78419 

78426 

78433 

78440 

78447 

78455 

609 

78462 

78469 

78476 

78483 

78  490 

78  497 

78504 

78512 

78519 

78  526 

61O 

78  533 

78540 

78  547 

78554 

78  561 

78569 

78576 

78583 

78590 

78  597 

611 

78604 

78611 

78618 

78  625 

78  633 

7S6HO 

78647 

78  654 

78661 

78668 

612 

78  675 

78682 

78  689 

78696 

78704 

78711 

78718 

78725 

78732 

78739 

613 

78746 

78  753 

78760 

78  767 

78774 

78  7S1 

78789 

78796 

78  803 

78810 

614 

78817 

78824 

78'S31 

78838 

78845 

78852 

78859 

78866 

78873 

78880 

615 

78888 

78895 

78902 

78909 

78916 

78923 

78930 

78937 

78944 

78951 

616 

78  958 

78  965 

78972 

78979 

78  986 

78  993 

79000 

79007 

79014 

79021 

617 

79029 

79036 

79  043 

79  050 

79  057 

79064 

79071 

79078 

79085 

79092 

618 

79099 

79106 

79  113 

79120 

79127 

79134 

79  141 

79148 

79  155 

79162 

619 

79169 

79176 

79183 

79190 

79197 

79204 

79211 

79218 

79  225 

79232 

62O 

79239 

79246 

79253 

79260 

79267 

79274 

79281 

79288 

79295 

79302 

621 

79309 

79316 

79323 

79330 

79  337 

79344 

79351 

79358 

79365 

79372 

622 

79379 

79386 

79393 

79400 

79  407 

79414 

79421 

79428 

79435 

79442 

623 

79449 

79  456 

79463 

79470 

79  477 

79  484 

79  491 

79498 

79  505 

79511 

624 

79518 

79525 

79532 

79539 

79546 

79553 

79  560 

79567 

79574 

79581 

625 

79  588 

79595 

79602 

79609 

79616 

79623 

79630 

79637 

79644 

79650 

626 

79657 

79664 

79671 

79678 

79  685 

79692 

79699 

79706 

79  713 

79720 

627 

79727 

79734 

79741 

79748 

79  754 

79761 

79768 

79  775 

79782 

79  789 

628 

79796 

79803 

79810 

79817 

79824 

79831 

79837 

79  844 

79851 

79  858 

629 

79865 

79872 

79879 

79886 

79  893 

79900 

79  906 

79913 

79920 

79927 

63O 

79934 

79941 

79948 

79955 

79962 

79969 

79975 

79982 

799S9 

79996 

631 

80003 

80010 

80017 

80024 

80030 

80037 

80044 

80051 

80058 

SO  065 

632 

80072 

80079 

80085 

80092 

80099 

80106 

80113 

80120 

80127 

80134 

633 

80140 

80147 

80154 

80161 

80168 

80175 

80182 

80188 

80195 

SO  202 

634 

80209 

80216 

80223 

80229 

80236 

80243 

80250 

80257 

SO  264 

SO  271 

635 

80277 

80284 

80291 

80298 

80305 

80312 

80  318 

80  325 

80  332 

80  339 

636 

80346 

80  353 

80359 

80  366 

80373 

80380 

80387 

80393 

80400 

80407 

637 

80414 

80421 

80428 

80  434 

80441 

80448 

80  455 

80462 

SO  468 

80  475 

638 

80482 

80489 

80496 

80502 

80509 

80516 

80  523 

80  530 

SO  536 

80  543 

639 

80550 

80557 

80564 

80570 

80577 

80  584 

80591 

80598 

80604 

SO  611 

640 

80618 

80625 

80632 

80638 

SO  645 

80652 

80  659 

80665 

SO  672 

80679 

641 

80686 

80693 

80699 

80706 

80713 

80720 

80786 

SO  733 

80740 

SO  747 

642 

80754 

80760 

80767 

80774 

80781 

80787 

80794 

80801 

80  SOS. 

SO  814 

643 

80821 

80828 

80835 

80841 

80848 

80855 

80862 

SOS6S 

SO  875 

80882 

644 

80889 

80895 

80902 

80909 

80916 

80922 

80929 

80936 

80943 

80949 

645 

80956 

80963 

80969 

80976 

80983 

80990 

80996 

SI  003 

81010 

81017 

646 

81023 

81030 

81037 

81  043 

81  050 

81057 

81064 

81  070 

81077 

SI  084 

647 

81090 

81097 

81  104 

81111 

81  117 

81  124 

81131 

81137 

81144 

SI  151 

648 

81  158 

81  164 

81171 

81178 

81  184 

81191 

81  198 

81204 

81211 

81218 

649 

81224 

81  231 

81238 

81245 

81251 

81258 

81265 

81271 

8127S 

81285 

No. 

O 

1 

2 

3 

4 

5 

G 

7 

8 

9 

600-649 


650-699 


21 


No. 

O          1           2          3          4 

50789 

65O 

651 
652 
653 
654 

81291    81298    81305    81311    81318 
81358    81365    81371    81378    81385 
81425    81431    81438    81445    81451 
81491    81498    81505    81511    81518 
81  558    81  564    81  571    81  578    81  584 

81  325    81  331    81  338    81  345    81  351 
81391    81398    81405    81411    81418 
81458    81465    81471    81478    81485 
81525    81531    81538    81544    81551 
81591    81598    81604    81611    81617 

655 
656 
657 
658 
659 

81624    81631    81637    81644    81651 
81  690    81  697    81  704    81  710    81  717 
81  757    81  763    81  770    81  776    81  783 
81823    81829    81836    81842    81849 
81889    81895    81902    81908    81915 

81  657    81  664    81  671    81  677    81  684 
81  723    81  730    81  737    81  743    81  750 
81  790    81  796    81  803    81  809    81  816 
81  856    81  862    81  869    81  875    81  882 
81921    81928    81935    81941    81948 

66O 

661 
662 
663 
664 

81954    81961    81968    81974    81981 
82020    82027    82033    82040    82046 
82  086    82092    82099    82105    82112 
82151    82-158    82164    82171    82178 
82217    82223    82230    82236    82243 

81987    81994    82000    82007    82014 
82053    82060    82066    82073    82079 
82119    82125    82132    82138    82145 
82184    82191    82197    82204    82210 
82249    82256    82263    82269    82276 

665 
666 
667 
668 
669 

82282    82289    82295    82302    82308 
82347    82354    82360    82367    82373 
82413    82419    82426    82432    82439 
82478    82484    82491    82497    82504 
82543    82549    82556    82562    82569 

82315    82321    82328    82334    82341 
82380    82387    82393    82400    82406 
82445    82452    82458    82465    82471 
82510    82517    82523    82530    82536 
82575    82582    82588    82595    82601 

67O 

671 
672 
673 
674 

82607    82614    82620    82627    82633 
82672    82679    82685    82692    82698 
82737    82743    82750    82756    82763 
82802    82808    82814    82821    82827 
82866    82872    82879    82885    82892 

82640    82646    82653    82659    82666 
82705    82711    82718    82724    82730 
82769    82776    82782    82789    82795 
82834    82840    82847    82853    82860 
82898    82905    82911    82918    82924 

675 
676 
677 
678 
679 

82930    82937    82943    82950    82956 
82995    83001    83008    83014    83020 
83059    83065    83072    83078    83085 
83  123    83  129    83  136    83  142    83  149 
83  187    83  193    83  200    83  206    83  213 

82963    82969    82975    82982    82988 
83027    83033    83040    83046    83052 
83  091    83  097    83  104    83  110    83  117 
83  155    83  161    83  168    83  174    83  181 
83219    83225    83232    83238    83245 

68O 

681 
682 
683 

684 

83251    83257    83264    83270    83276 
83315    83321    83327    83334    83340 
83378    83385    83391    83398    83404 
83442    83448    83455    83461    83467 
83506    83512    83518    83525    83531 

83283    83289    83296    83302    83308 
83347    83353    83359    83366    83372 
83410    83417    83423    83429    83436 
83474    83480    83487    83493    83499 
83537    83544    83550    83556    83563 

685 
686 
687 
688 
689 

83569    83575    83582    83588    83594 
83632    83639    83645    83651    83658 
S3  696    83  702    83  708    83  715    83  721 
83  759    83  765    83  771    83  778    83  784 
83822    83828    83835    83841    83847 

83601    83607    83613    83620    83626 
83664    83670    83677    83683    83689 
83727    83734    83740    83746    83753 
83790    83797    83803    83809    83816 
83853    83860    83866    83872    83879 

690 

691 
692 
693 
694 

83885    83891    83897    83904    83910 
83948    83954    83960    83967    83973 
84011    84017    84023    84029    84036 
84073    84080    84086    84092    84098 
84  136    84  142    84  148    84  155    84  161 

83916    83923    83929    83935    83942 
83979    83985    83992    83998    84004 
84042    84048    84055    84061    84067 
84105    84111    84117    84123    84130 
84167    84173    84180    84186    84  If  2 

695 
696 
697 
698 
699 

84198    84205    84211    84217    84223 
84261    84267    842^    84280    84286 
84323    84330    84336    84342    84348 
843S6    84392    84398    84404    84410 
84448    84454    84460    84466    84473 

84230    84236    84242    84248    A0d? 
84292    84298    84305    84311    84317 
84354    84361    84367    84373    84379 
84417    84423    84429    84435    84442 
84  479    84  485    84  491    84  497    84  504 

No. 

O           1           2           3          4 

56789 

650-699 


9,9, 


700-749 


No. 

O          1           2          3          4 

5           6           7           89 

700 

701 
702 
703 
704 

84510  84516  84522  84528  84535 
84572  84578  84584  84590  84597 
84634  84640  84646  84652  84658 
84696  84702  84708  84714  84720 
84757  84763  84770  84776  84782 

84541    84547    84553    84559    84566 
84603    84609    84615    84621    84628 
84665    84671    84677    84683    84689 
84726    84733    84739    84745    84751 
84788    84794    84800    84807    84813 

705 
706 
707 
708 
709 

84819  84825  84831  84837  84844 
84880  84887  84893  84899  84905 
84942  84948  84954  84960  84967 
85003  85009  85016  85022  85028 
85065  85071  85077  85083  85089 

84850    84856   84862    84868    84874 
84911    84917    84924    84930    84936 
84973    84979    84985    84991    84997 
85034    85040    85046    85052    85058 
85  095    85  101    85  107    85  114    85  120 

710 

711 
712 
713 
714 

85  126  85  132  85  138  85  144  85  150 
85  187  85  193  85  199  85  205  85  211 
85  248  85  254  85  260  85  266  85  272 
85309  85315  85321  85327  85333 
85370  85376  85382  85388  85394 

85156   85163    85169    85175    85181 
85  217    85  224    85  230    85  236    85  242 
85278    85285    85291    85297    85303 
85339    85345    85352    85358    85364 
85400    85406    85412    85418    85425 

715 
716 
717 
718 
719 

85431  85437  85443  85449  85455 
85  491  85  497  85  503  85  509  85  516 
85552  85558  85564  85570  85576 
85612  85618  85625  85631  85637 
85673  85679  85685  85691  85697 

85461    85467    85473    85479    85485 
85522    85528    85534    85540    85546 
85582    85588    85594    85600    85606 
85643    85649    85655    85661    85667 
85703    85709   85715    85721    85727 

72O 

721 

722 
723 
724 

85  733  85  739  85  745  85  751  85  757 
85794  85800  85806  85812  85818 
85854  85860  85866  85872  85878 
85914  85920  85926  85932  85938 
85974  85980  85986  85992  85998 

85  763    85  769    85  775    85  781    85  788 
85824    85830    85836    85842    85848 
85884    85890    85896    85902    85908 
85944    85950    85956    85962    85968 
86004    86010    86016    86022    86028 

725 
726 

727 
728 
729 

86034  86040  86046  86052  86058 
86094  86100  86106  86112  86118 
86153  86159  86165  86171  86177 
86213  86219  86225  86231  86237 
86273  86279  86285  86291  86297 

86064    86070    86076    86082    86088 
86124    86130    86136    86141    86147 
86183    86189    86195    86201    86207 
86243    86249    86255    86261    86267 
86303    86308    86314    86320    86326 

73O 

731 
732 
733 
734 

86332  86338  86344  86350  86356 
86392  86398  86404  86410  86415 
864S1  86457  86463  86469  86475 
86510  86516  86522  86528  86534 
86570  86576  86581  86587  86593 

86362    86368    86374    86380   86386 
86421    86427    86433    86439    86445 
86481    86487    86493    86499    86504 
86540    86546    86552    86558    86564 
86599   86605    86611    86617    86623 

735 
736 
737 
738 
739 

86629  86635  86641  86646  86652 
86688  86694  86700  86705  86711 
86747  86753  86759  86764  86770 
86806  86812  86817  86823  86829 
86864  86870  86876  86882  86888 

86658    86664    86670    86676    86682 
86717    86723    86729    86735    867-41 
86  776    86  782    86  788    86  794    86  800 
86835    86841    86847    86  853    86859 
86894    86900    86906    86911    86917 

74O 

741 
742 
743 
744 

86923  86929  86935  86941  86947 
86982  86988  86994  86999  87005 
87040  87046  87052  87058  87064 
87099  87105  87111  87116  87122 
87157  87163  87169  87175  87181 

86953    86958    86964    86970    86976 
87011    87017    87023    87029    87035 
87070    87075    87081    87087    87093 
87128    87134    87140    87146    87151 
87186    87192    87198    87204    87210 

745 
746 
747 
748 
749 

87216  87221  87227  87233  87239 
87274  87280  87286  87291  87297 
87332  87338  87344  87349  87355 
87390  87396  87402  87408  87413 
87448  87454  87460  87466  87471 

87245    87251    87256    87262    87268 
87303    87309    87315    87320    87326 
87361    87367    87373    87379    87384 
87419-  87425    87431    87437    87442 
87477    87483    87489    87495    87500 

No. 

O           1           2          3          4 

56789 

700-749 


750-799 


23 


No. 

O           1           2          3          4 

56789 

750 

751 
752 
753 
754 

87506    87512    87518    87523    87529 
87564    87570    S7~576    87581    87587 
87622    87628    87633    87639    87645 
87679    87685    87691    87697    87703 
87737    87743    87749    87754    87760 

87535    87541    87547    87552    87558 
87593    87599    87604    87610    87616 
87651    87656    87662    87668    87674 
87708    87714    87720    87726    87731 
87766    87772    87777    87783    87789 

755 
756 
757 
758 
759 

87795    87800    87806    87812    87818 
87  852    87858    87864    87869    87875 
87910    87915    87921    87927    87933 
87967    87973    87978    87984    87990 
88024    88030    88036    88041    88047 

87823    87829    87835    87841    87846 
87881    87887    87892    87898    87904 
87938    87944    87950    87955    87961 
87996    88001    88007    88013    88018 
88053    88058    88064    88070    88076 

76O 

761 
762 
763 
764 

SSOSl    88087    88093    88098    88104 
88138    88144    88150    88156    88161 
88195    88201    88207    88213    88218 
88252    88258    88264    88270    88275 
88  309    88315    88321    88326    88332 

88110   88116    88121    88127    88133 
88167    88173    88178    88184    88190 
88224    88230    88235    88241    88247 
88281    88287    88292    88298    88304 
88338    88343    88349    88355    88360 

765 
766 
767 
768 
769 

88366   88372    88377    88383    88389 
88423    88429    88434    88440    88446 
88480    88485    88491    88497    88502 
88536    88542    88547    88553    88559 
88593    88598    88604    88610    88615 

88395    88400    88406    88412    88417 
88451    88457    88463    88468    88474 
88508    88513    88519    88525    88530 
88564    88570    88576    88581    88587 
88621    88627    88632    88638    88643 

770 

771 

772 
773 
774 

88649   88655    88660   88666    88672 
88705    88711    88717    88722    88728 
88762    88767    88773    88779    88784 
88818    88824    88829    88835    88840 
88874    88880    88885    88891    88897 

88677    88683    88689    88694    88700 
88734    88739    88745    88750    88756 
88790    88795    88801    88807    88812 
88846    88852    88857    88863    88868 
88902    88908    88913    88919    88925 

775 
776 
777 
778 
779 

88930    88936    88941    88947    88953 
88986    88992    88997    89003    89009 
-89042    89048    89053    89059    89064 
89098    89104    89109    89115    89120 
89154    89159    89165    89170    89176 

88958    88964    88969    88975    88981 
89014    89020    89025    89031    89037 
89070    89076    89081    89087    89092 
89126    89131    89137    89143    89148 
89182    89187    89193    89198    89204 

780 

781 
782 
783 
784 

89209    89215    89221    89226    89232 
89265    89271    89276    89282    89287 
89321'  89326    89332    89337    89343 
89376    89382    89387    89393    89398 
89432    89437    89443    89448    89454 

89237    89243    89248    89254    89260 
89293    89298    89304    89310    89315 
89348    89354    89360    89365    89371 
89404    89409    89415    89421    89426 
89459    89465    89470    89476    89481 

785 
786 
787 
788 
789 

89487    89492    89498    89504    89509 
89542    89548    89553    89559    89564 
89597    89603    89609    89614    89620 
89653    89658    89664    89669    89675 
89708    89713    89719    89724    89730 

89515    89520    89526    89531    89537 
89570   89575    89581    89586    89592 
89625    89631    89636    89642    8^647 
89680   89686    89691    89697    89702 
89735    89741    89746    89752    89757 

790 

791 
792 
793 
794 

89763    89768    89774    89779    89785 
89  818    89823    89829    89834    89840 
89873    89878    89883    89889    89894 
89927    89933    89938    89944    89949 
89982    89988    89993    89998    90004 

89790    89796    89801    89807    89812 
89845    89851    89856    89862    89867 
89900    89905    89911    89916    89922 
89955    89960    89966    89971    89977 
90009    90015    90020    90026    90031 

795 
796 
797 
798 
799 

90037    90042    90048    90053    90059 
90091    90097    90102    90108    90113 
90146    90151    90157    90162    90168 
90200    90206    90211    90217    90222 
90255    90260    90266    90271    90276 

90064    90069    90075    90080    90086 
90119    90124    90129    90135    90140 
90173    90179    90184    90189    90195 
90227    90233    90238    90244    90249 
90282    90287    90293    90298    90304 

No. 

O           1           2           3          4 

5          G           7          8          9 

750-799 


800-849 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

80O 

90309 

90314 

90320 

90  325 

90331 

90  336 

90342 

90347 

90  352 

90  35S 

801 

90363 

90369 

90374 

90380 

90385 

90390 

90  396 

90401 

90  407 

90412 

802 

90417 

90423 

90428 

90  434 

90  439 

90445 

90  450 

90  455 

90  461 

90466 

803 

90  472 

90  477 

90  4  82 

904SS 

90493 

90  499 

90504 

90  509 

90515 

90520 

804 

90526 

90531 

90536 

90542 

90547 

90553 

90558 

90563 

90569 

90574 

805 

90  580 

90585 

90590 

90596 

90601 

90607 

90  612 

90617 

90623 

90628 

806 

90  634 

90  639 

90  644 

90650 

90  655 

90660 

90  666 

90671 

90  677 

90  682 

807 

90687 

90693 

90698 

90703 

90709 

90714 

90720 

90  725 

90  730 

90  736 

808 

90741 

90747 

90752 

90757 

90763 

90768 

90773 

90779 

90  7S4 

90  7  89 

809 

90795 

90800 

90806 

90811 

90  816 

90822 

90827 

90  832 

90  838 

90843 

810 

90849 

90854 

90  859 

90865 

90S70 

90875 

90  SSI 

90  886 

90891 

90  S97 

811 

90902 

90  907 

90  913 

90918 

90924 

90  929 

90934 

90  940 

90  945 

90  950 

812 

90956 

90961 

90966 

90  972 

90  977 

90  982 

90  9SS 

90  993 

90  998 

91  004 

813 

91009 

91014 

91020 

91025 

91030 

91036 

91041 

91  046 

91  052 

91  057 

814 

91062 

91068 

91073 

91  078 

91084 

91  OS9 

91094 

91100 

91  105 

91  110 

815 

91  116 

91  121 

91  126 

91  132 

91  137 

91  142 

91  148 

91  153 

91  158 

91  164 

816 

91  169  . 

91  174 

91  ISO 

91  185 

91  190 

91  196 

91  201 

91  206 

91212 

91217 

817 

91222 

91228 

91233 

91238 

91243 

91  249 

91  254 

91  259 

91  265 

91270 

818 

91275 

9128L 

91  286 

91  291 

91297 

91302 

91307 

91  312 

91318 

91323 

819 

91328 

91334 

91339 

91344 

91350 

91355 

91360 

91365 

91371 

91376 

82O 

91381 

91387 

91  392 

91397 

91403 

91  408 

91413 

91418 

91424 

91429 

821 

91434 

91440 

91445 

91450 

91  455 

91461 

91  466 

91  471 

91477 

91  482 

822 

91487 

91492 

91  498 

91  503 

91  508 

91514 

91519 

91  524 

91  529 

91535 

823 

91540 

91  545 

91  551 

91556 

91  561 

91566 

91  572 

91  577 

91  582 

91  5S7 

824 

91  593 

91  598 

91603 

91609 

91614 

91  619 

91624 

91  630 

91  635 

91  640 

825 

91645 

91651 

91656 

91661 

91666 

91672 

91677 

91  682 

91687 

91693 

826 

91698 

91  703 

91  709 

91714 

91  719 

91  724 

91  730 

91735 

91  740 

91  745 

827 

91751 

91  756 

91  761 

91766 

91772 

91  777 

91  782 

91  787 

91  793 

91798 

828 

91803 

91808 

91814 

91  819 

91S24 

91S29 

91  834 

91  840 

91  845 

91  850 

829 

91855 

91  861 

91866 

91871 

91876 

91  882 

91SS7 

91892 

91  S97 

91  903 

830 

91  90S 

91913 

91918 

91924 

91929 

91934 

91939 

91944 

91  950 

91  955 

831 

91960 

91965 

91971 

91976 

91  981 

91  986 

91  991 

91  997 

92002 

92  007 

832 

92012 

92018 

92023 

92028 

92  033 

92  038 

92044 

92049 

92  054 

92  059 

833 

92065 

92070 

92075 

920SO 

92  085 

92  091 

92  096 

92101 

92  106 

92111 

834 

92117' 

92122 

92  127 

92  132 

92137 

92143 

92  148 

92153 

92158 

92163 

835 

92169 

92174 

92179 

92  184 

921S9 

92195 

92200 

92205 

92210 

92215 

836 

92221 

92226 

92231 

92236 

92  2-11 

92  247 

92  252 

92  257 

92  262 

92267 

837 

92  273 

92278 

92  283 

922SS 

92293 

92298 

92  304 

92  309 

92314 

92319 

838 

92324 

92330 

92335 

92340 

92345 

92  350 

92355 

92361 

92366 

92371 

839 

92376 

92381 

92387 

92392 

92397 

92402 

92407 

92412 

92418 

92423 

840 

92428 

92433 

92  438 

92443 

92449 

92  454 

92  459 

92464 

92469 

92474 

841 

92480 

92485 

92  490 

92495 

92  500 

92  505 

92  511 

92  516 

92  521 

92526 

842 

92531 

92536 

92542 

92  547 

92  552 

92  557 

92  562 

92567 

92572 

92  5  78 

843 

92583 

92588 

92593 

92598 

92  603 

92  609 

92614 

92619 

92624 

92629 

844 

92634 

92639 

92645 

92650 

92655 

92660 

92665 

92670 

92675 

92681 

845 

92686 

92691 

92696 

92701 

92706 

92711 

92716 

92722 

92727 

92  732 

846 

92  737 

92742 

92747 

92  752, 

92  758 

92763 

92  768 

92773 

92  778 

92  783 

847 

92788 

92  793 

92799 

92804 

92809 

92  814 

92  819 

92824 

92  829 

92  834 

848 

92840 

92  845 

92850 

92855 

92  860 

92  865 

92  870 

92  875 

92  SSI 

92SS6 

849 

92891 

92896 

92901 

92906 

92911 

92916 

92921 

92927 

92  932 

92  937 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

800-849 


850-899 


25 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

850 

92942 

92947 

92952 

92  957 

92962 

92967 

92973 

92978 

92983 

92988 

851 

92  993 

92  Si98 

93003 

93008 

93013 

93018 

93024 

93029 

93034 

93039 

852 

93044 

93049 

93  054 

93  059 

93064 

93069 

93075 

93080 

93085 

93090 

853 

93095 

93  100 

93105 

93110 

93115 

93120 

93125 

93131 

93136 

93  141 

854 

93146 

93151 

93156 

93161 

93166 

93171 

93176 

93181 

93186 

93192 

855 

93197 

93202 

93207 

93212 

93217 

93222 

93227 

93232 

93237 

93242 

856 

93247 

93252 

93  258 

93263 

93268 

93  273 

93278 

93283 

93  288 

93293 

857 

93  298 

93303 

93308 

93313 

93  318 

93323 

93328 

93334 

93339 

93344 

858 

93  349 

93  354 

93  359 

93364 

93369 

93374 

93379 

93384 

93389 

93394 

859 

93  399 

93404 

93409 

93414 

93420 

93425 

93430 

93435 

93440 

93445 

860 

93450 

93  455 

93460 

93  465 

93470 

93475 

93480 

93485 

93490 

93495 

861 

93  500 

93  505 

93510 

93  515 

93  520 

93  526 

93531 

93536 

93  541 

93546 

862 

93551 

93  556 

93  561 

93566 

93571 

93576 

93  581 

93  586 

93591 

93596 

863 

93601 

93606 

93611 

93616 

93621 

93626 

936.31 

93636 

93641 

93646 

864 

93651 

93656 

93661 

93666 

93671 

93676 

93682 

93687 

93692 

93697 

865 

93702 

93707 

93712 

93717 

93722 

93727 

93732 

93737 

93742 

93747 

866 

93  752 

93  757 

93762 

93767 

93772 

93777 

93782 

93787 

93792 

93797 

867 

93802 

93807 

93812 

93817 

93822 

93827 

93832 

93837 

93842 

93847 

868 

93852 

93857 

93862 

93867 

93872 

93877 

93882 

93887 

93892 

93897 

869 

93902 

93907 

93912 

93917 

93922 

93927 

93932 

93937 

93942 

93947 

870 

93952 

93957 

93962 

93967 

93972 

93977 

93982 

93987 

93992 

93997 

871 

94002 

94007 

94012 

94017 

94022 

94027 

94032 

94037 

94042 

94047 

872 

94  052 

94057 

94  062 

94067 

94072 

94077 

94082 

94086 

94091 

94096 

873 

94101 

94106 

94111 

94116 

94121 

94126 

94131 

94136 

94141 

94146 

874 

94151 

94156 

94161 

94166 

94171 

94176 

94181 

94186 

94191 

94196 

875 

94201 

94206 

94211 

94216 

94221 

94226 

94231 

94236 

94240 

94245 

876 

94  250 

94255 

94260 

94265 

94270 

94275 

94280 

94285 

94290 

94295 

877 

94  300 

94305 

94310 

94315 

94320 

94325 

94330 

94335 

94340 

94345 

878 

94  349 

94354 

94359 

94364 

94369 

94374 

94379 

94384 

94389 

94394 

879 

94399 

94404 

94409 

94414 

94419 

94424 

94429 

94433 

94438 

94443 

88O 

94448 

94453 

94  458 

94463 

94468 

94473 

94478 

94483 

94488 

94493 

881 

94498 

94  503 

94507 

94512 

94517 

94522 

94527 

94532 

94537 

94542 

882 

94  547 

94  552 

94557 

94  562 

94  567 

94571 

94576 

94581 

94586 

94591 

883 

94596 

94601 

94606 

94611 

94616 

94621 

94626 

94630 

94  635 

94640 

884 

94645 

94650 

94655 

94660 

94665 

94670 

94675 

94680 

94685 

94689 

885 

94694 

94699 

94  704 

94709 

94714 

94719 

94724 

94729 

94734 

94738 

886 

94743 

94748 

94753 

94  758 

94763 

94768 

94773 

94778 

94783 

94787 

887 

94792 

94797 

94802 

94  807 

94812 

94817 

94822 

94827 

94832 

94836 

888 

94841 

94846 

94851 

94856 

94861 

94866 

94871 

94876 

94880 

94  885 

889 

94890 

94895 

94900 

94905 

94910 

94915 

94919 

94924 

94929 

94934 

89O 

94939 

94944 

94949 

94954 

94  959 

94963 

94968 

94973 

94978 

94983 

891 

94988 

94993 

94998 

95002 

95007 

95  012 

95  017 

95022 

95  027 

95  032 

892 

95036 

95041 

95046 

95051 

95  056 

95061 

95  066 

95071 

95075 

95080 

893 

95085 

95090 

95095 

95  100 

95  105 

95109 

95  114 

95119 

95  124 

95  129 

894 

95134 

95139 

95143 

95148 

95153 

95  158 

95163 

95168 

95173 

95177 

895 

95  182 

95187 

95192 

95197 

95202 

95  207 

95211 

95216 

95221 

95  226 

896 

95  231 

95  236 

95  240 

95  245 

95  250 

95  255 

95260 

95265 

95  270 

95  274 

897 

95279 

95284 

95289 

95294 

95299 

95  303 

95308 

95313 

95318 

95323 

898 

95328 

95332 

95337 

95342 

95  347 

95  352 

95357 

95361 

95  366 

95  371 

899 

95376 

95381 

95386 

95390 

95395 

95400 

95405 

95410 

95415 

95419 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

850-899 


26 


900-949 


No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9OO 

95424 

95429 

95434 

95439 

95444 

95448 

95  453 

95458 

95463 

95468 

901 

95472 

95477 

95  482 

95487 

95492 

95497 

95501 

95  406 

95511 

95516 

902 

95  521 

95  525 

95530 

95535 

95  540 

95545 

95  550 

95  554 

95559 

95564 

903 

95569 

95574 

95578 

95  583 

95588 

95  593 

95  598 

95602 

95607 

95612 

904 

95617 

95622 

95626 

95631 

95636 

95641 

95646 

95650 

95655 

95660 

905 

95  665 

95670 

95674 

95679 

95684 

95689 

95694 

95698 

95703 

95708 

906 

95713 

95718 

95  722 

95727 

95732 

95737 

95  742 

95746 

95751 

95  756 

907 

95761 

95  766 

95770 

95775 

95  780 

95785 

95789 

95794 

95799 

95804 

908 

95809 

95813 

95818 

95823 

95828 

95832 

95837 

95  842 

95  847 

95  852 

909 

95856 

95861 

95866 

95871 

95875 

95880 

95885 

95890 

95895 

95899 

910 

95904 

95909 

95  914 

95918 

95923 

95928 

95933 

95  938 

95942 

95947 

911 

95952 

95957 

95961 

95966 

95  971 

95976 

95980 

95985 

95990 

95995 

912 

95999 

96004 

96009 

96014 

96019 

96023 

96028 

96033 

96  038 

96042 

913 

96  047 

96052 

96057 

96061 

96  066 

96071 

96076 

96080 

96085 

96090 

914 

96095 

96099 

96104 

96109 

96114 

96118 

96123 

96128 

96133 

96137 

915 

96142 

96147 

96152 

96  156 

96161 

96166 

96171 

96175 

96180 

96185 

916 

96  190 

96194 

96  199 

96204 

96209 

96213 

96218 

96223 

96227 

96232 

917 

96237 

96242 

96246 

96251 

96  256 

96  261 

96265 

96270 

96275 

96280 

918 

96284 

96289 

96294 

96298 

96303 

96308 

96313 

96317 

96322 

96327 

919 

96332 

96336 

96341 

96346 

96350 

96355 

96360 

96365 

96369 

96374 

92O 

96379 

96384 

96388 

96393 

96398 

96402 

96407 

96412 

96417 

96421 

921 

96426 

96431 

96435 

96440 

96445 

96450 

96454 

96459 

96464 

96468 

922 

96473 

96478 

96483 

96487 

96492 

96497 

96501 

96506 

96511 

96515 

923 

96520 

96  525 

96530 

96  534 

96539 

96  544 

96  548 

96  553 

96558 

96562 

924 

96567 

96572 

96577 

96581 

96586 

96591 

96595 

96600 

96605 

96  609 

925 

96614 

96619 

96624 

96628 

96633 

96638 

96642 

96647 

96652 

96656 

926 

96661 

96666 

96670 

96675 

96680 

96685 

96689 

96694 

96699 

96703 

927 

96708 

96713 

96717 

96722 

96727 

96731 

'96736 

96741 

96745 

96750 

928 

96755 

96759 

96764 

96  769 

96774 

96778 

96783 

96788 

96792 

96797 

929 

96802 

96806 

96811 

96  816 

96820 

96825 

96830 

96834 

96839 

96844 

93O 

96848 

96853 

96858 

96862 

96867 

96872 

96876 

96881 

96886 

96890 

931 

96895 

96900 

96904 

96909 

96914 

96918 

96923 

96928 

96  932 

96937 

932 

96942 

96946 

96951 

96956 

96  960 

96965 

96  970 

96974 

96979 

96984 

933 

96988 

96993 

96997 

97002 

97007 

97011 

97016 

97021 

97025 

97030 

934 

97035 

97039 

97044 

97049 

97053 

97058 

97063 

97067 

97072 

97077 

935 

97081 

97086 

97090 

97  095 

97100 

97104 

97109 

97114 

97118 

97123 

936 

97128 

97132 

97137 

97142 

97146 

97151 

97  155 

97160 

97165 

97169 

937 

97174 

97179 

97183 

97188 

97192 

97197 

97202 

97206 

97211 

97216 

938 

97220 

97225 

97230 

97234 

97239 

97243 

97248 

97253 

97257 

97262 

939 

97267 

97271 

97276 

97280 

97285 

97290 

97294 

97299 

97304 

97308 

940 

97313 

97317 

97322 

97327 

97331 

97336 

97340 

97345 

97350 

97354 

941 

97359 

97364 

97368 

97373 

97377 

97382 

97387 

97391 

97396 

97400 

942 

97405 

97410 

97414 

97419 

97424 

97428 

97  433 

97437 

97442 

97  447 

943 

97451 

97  456 

97  460 

97465 

97470 

97474 

97479 

97483 

97488 

97493 

944 

97497 

97  502 

97506 

97511 

97516 

97  520 

97  525 

97529 

97534 

97539 

945 

97543 

97548 

97  552 

97557 

97562 

97566 

9757,1 

97575 

97580 

97585 

946 

97  589 

97594 

97598 

97603 

97607 

97612 

97617 

97621 

97626 

97630 

947 

97  635 

97640 

97  644 

97649 

97653 

97658 

97663 

97667 

97672 

97676 

948 

97681 

97685 

97690 

97695 

.  97  699 

97704 

97708 

97713 

97717 

97722 

949 

97727 

97731 

97736 

97740 

97745 

97749 

97  754 

97759 

97763 

97768 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900-949 


950-1000 


27 


No. 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950 

97772 

97777 

97782 

97  786 

97791 

97795 

97800 

97804 

97  809 

97813 

951 

97818 

97823 

97827 

97  832 

97836 

97841 

97  845 

97850 

97855 

97859 

952 

97864 

97868 

97873 

97877 

97882 

97886 

97891 

97896 

97900 

97905 

953 

97909 

97  914 

97918 

97923 

97928 

97932 

97937 

97941 

97946 

97950 

954 

97955. 

97959 

97964 

97968 

97973 

97978 

97982 

97987 

97991 

97996 

955 

98000 

98005 

98009 

98014 

98019 

98023 

98028 

98032 

98037 

98041 

956 

98046 

98050 

98055 

98059 

98064 

98068 

98073 

98078 

98082 

98087 

957 

98091 

98096 

98100 

98105 

98109 

98114 

98118 

98123 

98127 

98132 

958 

98137 

98141 

98146 

98150 

98155 

98  159 

98164 

98  168 

98173 

98177 

959 

98182 

98186 

98191 

98195 

98200 

98204 

98209 

98214 

98218 

98223 

96O 

98227 

98  232 

98  236 

98241 

98  245 

98250 

98254 

98259 

98263 

98268 

961  98272 

98277 

98281 

98286 

98290 

98295 

98299 

98304 

98308 

98313 

962  98318 

98322 

98327 

98331 

98336 

98340 

98345 

98349 

98354 

98358 

963  98363 

98367 

98372 

98376 

98  381 

98385 

98390 

98394 

98399 

98403 

964 

98408 

98412 

98417 

98421 

98426 

98430 

98435 

98439 

98444 

98448 

965 

98453 

98457 

98462 

98466 

98471 

98475 

98480 

98484 

98489 

98493 

966 

98498 

98502 

98507 

98511 

98516 

98520 

98525 

98529 

98534 

98538 

967 

98  543 

98547 

98552 

98556 

98561 

98565 

98570 

98574 

98579 

98  583 

968 

98588 

98592 

98597 

98601 

98605 

98610 

98614 

98  619 

98623 

98628 

969 

98632 

98637 

98641 

98646 

98650 

98  655 

98659 

98664 

98668 

98673 

97O 

98677 

98682 

98686 

98691 

98695 

98700 

98704 

98709 

98713 

98717 

971 

98722 

98726 

98731 

98735 

98740 

98744 

98749 

98753 

98758 

98762 

972 

98767 

98771 

98776 

98780 

98  784  . 

98789 

98793 

98798 

98802 

98807 

973 

98811 

98816 

98820 

98825 

98829 

98834 

98838 

98843 

98847 

98851 

974 

98856 

98860 

98865 

98869 

98874 

98878 

98883 

98887 

98892 

98896 

975 

98900 

98905 

98909 

98914 

98918 

98923 

98927 

98932 

98936 

98941 

976 

98945 

98949 

98954 

98958 

98963 

98967 

98972 

98976 

98981 

98985 

977 

98989 

98  994 

98998 

99003 

99007 

99012 

99016 

99021 

99025 

99029 

978 

99034 

99038 

99043 

99047 

99052 

99056 

99061 

99065 

99  069 

99074 

979 

99078 

99083 

99087 

99092 

99096 

99100 

99105 

99  109 

99114 

99118 

980 

99123 

99127 

99131 

99136 

99  140 

99  145 

99149 

99154 

99158 

99162 

981 

99167 

99171 

99176 

99  180 

99185 

99189 

99193 

99198 

99202 

99207 

982 

99211 

99216 

99220 

99224 

99  229 

99233 

99238 

99242 

99247 

99251 

983 

99255 

99260 

99264 

99269 

99273 

99277 

99282 

99286 

99291 

99  295 

984 

99300 

99304 

99308 

99313 

99317 

99322 

99326 

99330 

99335 

99339 

985 

99  344 

99348 

99352 

99357 

99361 

99366 

99370 

99374 

99379 

99383 

986 

99  388 

99392 

99396 

99401 

99405 

99410 

99414 

99419 

99423 

99427 

987 

99432 

99436 

99441 

99445 

99449 

99454 

99458 

99463 

99467 

99471 

988  99476 

99480 

99  484 

99489 

99493 

99498 

99  502 

99  506 

99511 

99515 

989  99520 

99524 

99528 

99533 

99537 

99542 

99546 

99550 

99  555 

99559 

990 

99  564 

99568 

99  572 

99  577 

99  581 

99  585 

99590 

99594 

99599 

99603 

991 

99607 

99612 

99616 

99621 

99625 

99629 

99  634 

99638 

99642 

99647 

992 

99651 

99656 

99660 

99664 

99  669 

99673 

99677 

99682 

99686 

99691 

993 

99695 

99699 

99704 

99708 

99712 

99717 

99721 

99726 

99730 

99734 

994  99739 

99743 

99747 

99752 

99756 

99760 

99765 

99769 

99774 

99778 

995 

99782 

99787 

99791 

99795 

99800 

99804 

99808 

99813 

99817 

99822 

996  99826 

99  830 

99835 

99839 

99843 

99848 

99  852 

99  856 

99861 

99865 

997  99870 

99874 

99878 

99883 

99887 

99891 

99896 

99900 

99904 

99909 

998  99913 

99917 

99922 

99926 

99930 

99935 

99939 

99944 

99948 

99952 

999 

99957 

99961 

99965 

99970 

99974 

99978 

99983 

99987 

99991 

99996 

1OOO  00000 

00004 

00009 

00013 

00017 

00022 

00026 

00030 

00035 

00039 

No.    0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950-1000 


TABLE   II -USEFUL  CONSTANTS  AND  THEIR  LOGARITHMS 


LOG 

Circumference  of  the  Circle  in  Degrees    —            360 

2.  55  630  250 

Circumference  of  the  Circle  in  Minutes    =       21  600 

4.3344537.S 

Circumference  of  the  Circle  in  Seconds     —1296000 

6.11  260500 

If  the  radius  =  1,  the  semi-circumference  is 

TT  —  3.  14  159  265  358  979  323  846  764  338  378 

0.49714987 

ALSO 

LOG 

7r2  =  9.86960440 

0.99429975 

27r=    6.28318531 

0.79817987 

—  =  0.10132118 

9.00570025  -  10 

47r  =  12.56637061 

1.09920986 

7T- 

—  =    1.57079633 

0.19611988 

VV  =  1.  77  245  385 

0.24857494 

2 

—  =    1.04719755 

0.02002862 

JL  =0.56418958 

9.  75  142  506  -  10 

3 

VTT 

~=   4.18879020 

0.  62  208  861 

/I 

3 

\/-  =  0.97  720  502 

9.98998569-  10 

—  =    0.78539816 

9.89508988-  10 

4 

/4 

\/-=  1.12837917 

0.  05  245  506 

—  =    0.52359878 

9.71899862-  10 

\7T 

6 

i 

-v/TT  =  1.46459189 

0.16571662 

-i.=    0.31830989 

9.  50  285  013  -  10 

7T 
1 

-^  =  0.68278406 

9.83428338-10 

—  =    0.  15  915  494 

9.20182013-10 

VTT 

2-jr 

—  =    0.95492966 

9.97997138-  10 

vV  =  2.  14  502  940 

0.33143325 

7T 

8/~5~" 

^-=    1.27323954 

0.10491012 

V/A.=  0.62  035  049 
\4w 

9.79263713  -  10 

7T 

—  =    0.  23  873  241 

9.  37  791  139  -  10 

•\8/—  =  0.80  599  598 

9.90633287-  10 

4?r 

\   6 

Angle  0,  whose  arc  is  equal  to  the  radius  r,  is 

180 
in  degrees,  0°  =                =57.29577951°   

7T 

1.75812263 

10800 

in  minutes,  0'   —                —3437  74677' 

3.53627388 

7T 

648000 

in  seconds    0"                         206264806" 

5.31  442  513 

Angle  2  0,  whose  arc  is  equal  to  twice  the  radius,  2  r,  is 

in  degrees,  20°=       —       =  114.  59  155  903° 

2.  05  915  263 

7T 

in  minutes,  2  0'   -    21  60°    -687549354' 

3.837303*88 

7T 

in  seconds  20"      1296000      412529612" 

5.61545513 

7T 

If  the  radius  r  =  1,  the  length  of  the  arc  is  : 

for  1  degree  =  -~-  =       —       =  0.  01  745  329     . 

8.24187737-10 

0J             180 

for  1  minute  =  -L  =    —  ?_    =  0.  00  029  089     , 

6.46372612-  10 

0'           10  800 

for  1  second  =  -L-  =   —  -  —    =  0.  00  000  485 

4.  68  557  487  ~  10 

0"         648000 

for  &  degree  =  -^  =       ^       =0.00872665     .... 

7.94084737-  10 

for  2  minute  - 

1              TT           c 

1.00014544     .... 

6.16269612      10 

2  0'         21  600 

for  .>  second  - 

1                «r             - 

.00000242     .... 

4.38454487      10 

20"      1296000 

Sin  1",  when  the  radius  r  -  1,  is  .     .     .=0.00000485     .     .     .     . 

4.68557487-  10 

TABLE  III 

LOGARITHMS 

OF    THE 

TRIGONOMETRIC    FUNCTIONS 

From  0°  0'  to  0°  3',  and  from  89°  57'  to  90°,  for  every  second 
From  0°  to  2°,  and  from  88°  to  90°,  for  every  ten  seconds 
From  1°  to  89°,  for  every  minute 

NOTE.  —  The  characteristic  of  every  logarithm  in  the  following  table  is  too 
large  by  10.     Therefore,  —  10  should  be  written  after  every  logarithm. 

L  sin  and  L  tan                     U                       I<  sin  and  L  tan 

//. 

0 

1' 

6.46  373 
6.47  090 
6.47  797 
6.48  492 
6.49  175 

2' 

6.76476 
6.76  836 
6.77  193 
6.77  548 
6.77  900 

// 

// 

O' 

6.16270 
6.17694 
6.19072 
6.20  409 
6.21  705 

1' 

6.63  982 
6.64  462 
6.64  936 
6.65  406 
6.65  870 

2' 

6.86  167 
6.86455 
6.86  742 
6.87  027 
6.87  310 

// 

O 

1 

2 

3 
4 

6O 

59 

58 
57 
56 

30 

31 
32 
33 
34 

30 

29 
28 
27 
26 

4.68  557 
4.98  660 
5.16270 
5.28  763 

5 
6 

7 
8 
9 

5.38  454 
5.46373 
5.53  067 
5.58  866 
5.63  982 

6.49  849 
6.50512 
6.51  165 
6.51  808 
6.52  442 

6.78  248 
6.78  595 
6.78  938 
6.79278 
6.79616 

55 
54 
53 
52 
51 

35 
36 
37 
38 
39 

6.22  964 
6.24  188 
6.25  378 
6.26  536 
6.27  664 

6.66  330 
6.66  785 
6.67  235 
6.67  680 
6.68  121 

6.87  591 
6.87  870 
6.88  147 
6.88  423 
6.88  697 

25 
24 
23 
22 
21 

10 

11 
12 
13 
14 

5.68557 
5.72  697 
5.76476 
5.79952 
5.83  170 

6.53  067 
6  53  683 
6.54  291 
6.54  890 
6.55  481 

6.79952 
6.80  285 
6.80615 
6.80  943 
6.81  268 

50 

49 
48 
47 
46 

40 

41 
42 
43 
44 

6.28  763 
6.29  836 
6.30  882 
6.31  904 
6.32  903 

6.68  557 
6.68  990 
6.69  418 
6.69  841 
6.70  261 

6.88  969 
6.89  240 
6.89  509 
6.89  776 
6.90  042 

2O 

19 
18 
17 
16 

15 
16 
17 
18 
19 

5.86  167 
5.88969 
5.91  602 
5.94085 
5.96433 

6.56064 
6.56  639 
6.57  207 
6.57  767 
6.58  320 

6.81  591 
6.81  911 
6.82  230 
6.82  545 
6.82  859 

45 
44 
43 
42 
41 

45 
46 
47 
48 
49 

6.33  879 
6.34  833 
6.35  767 
6.36  682 
6.37  577 

6.70676 
6.71  088 
6.71  496 
6.71  900 
6.72  300 

6.90306 
6.90  568 
6.90  829 
6.91  088 
6.91346 

15 
14 
13 
12 
11 

20 

21 
22 
23 
24 

5.98660 
6.00  779 
6.02  800 
6.04  730 
6.06579 

6.58  866 
6.59  406 
6.59  939 
6.60  465 
6.60  985 

6.83  170 
6.83479 
6.83  786 
6.84  091 
6.84394 

40 

39 

38 
37 
66 

50 

51 

52 
53 
54 

6.38  454 
6.39315 
6.40  158 
6.40  985 
6.41  797 

6.72  697 
6.73  090 
6.73  479 
6.73  865 
6.74  248 

6.91  602 
6.91  857 
6.92  110 
6.92  362 
6.92  612 

1O 

9 

8 
7 
6 

25 
26 

27 
28 
29 

6.08351 
6.10055 
6.11694 
6.13  273 
6.14  797 

6.61  499 
6.62  007 
6.62  509 
6.63  006 
6.63  496 

6.84  694 
6.84  993 
6.85  289 
6.85  584 
6.85  876 

35 
34 
33 
32 
31 

55 
56 
57 
58 
59 

6.42  594 
6.43  376 
6.44  145 
6.44  900 
6.45  643 

6.74  627 
6.75  003 
6.75  376 
6.75  746 
6.76  112 

6.92  861 
6.93  109 
6.93  355 
6.93  599 
6.93  843 

5 
4 
3 

2 
1 

30 

6.16270 
59' 

6.63  982 
58' 

6.86  167 

57' 

30 

6O 

6.46373 
59' 

6.76  476 
58' 

6.94  085 

57' 

O 

" 

" 

" 

" 

L  cos  and  L  cot 


89' 


L  cos  and  L  cot 


29 


30 


0 


/    // 

L  sin      L  tan       L  cos 

//      / 

/    // 

L  sin      L  tan       I  cos 

//    / 

On 

10  00000 

0  fiO 

1O     0 

746^7^    7  46  37"?    1000000 

0  f»ft 

V 

10 

5.68557  5.68557  10.00000 

V/    \J  VJ 

50 

J_  Vr         \J 

10 

/  .  I  O  O  t  O      /  .  i  O  O  /  O      ±\J.\J\J  \J\J\J 

7.47090  7.47091   10.00000 

U  Ovr 

50 

20 

5.98660  5.98660  10.00000 

40 

20 

7.47797  7.47797  10.00000 

40 

30 

6.16270  6.16270  10.00000 

30 

30 

7.48491  7.48492  10.00000 

30 

40 

6.28763  6.28763  10.00000 

20 

40 

7.49175  7.49175   10.00000 

20 

50 

6.38454  6.38454  10.00000 

10 

50 

7.49  849  7.49849  10.00000 

10 

1    0 

6.46373  6.46373   10.00000 

0  59 

11     0 

7.50512  7.50512  10.00000 

049 

10 

6.53067  6.53067  10.00000 

50 

10 

7.51165   7.51165   10.00000 

50 

20 

6.58866  6.58866  10.00000 

40 

20 

7.51808  7.51809  10.00000 

40 

30 

6.63982  6.63982  10.00000 

30 

30 

7.52442  7.52443   10.00000 

30 

40 

6.68557  6.68557  10.00000 

20 

40 

7.53067  7.53067   10.00000 

20 

50 

6.72697  6.72697  10.00000 

10 

50 

7.53683  7.53683  10.00000 

10 

2   0 

6.76476  6.76476  10.00000 

0  58 

12    0 

7.54291  7.54291  10.00000 

048 

10 

6.79952  6.79952   10.00000 

50 

10 

7.54890  7.54890  10.00000 

50 

20 

6.83170  6.83170  10.00000 

40 

20 

7.55481   7.55481   10.00000 

40 

30 

6.86167  6.86167   10.00000 

30 

30 

7.56064  7.56064  10.00000 

30 

40 

6.88969  6.88969  10.00000 

20 

40 

7.56639  7.56639  10.00000 

20 

50 

6.91602  6.91602  10.00000 

10 

50 

7.57206  7.57206  10.00000 

10 

3   0 

6.94085  6.94085   10.00000 

0  57 

13    0 

7.57767  7.57767  10.00000 

047 

10 

6.96433  6.96433   10.00000 

50 

10 

7.58320  7.58320  10.00000 

50 

20 

6.98660  6.98661   10.00000 

40 

20 

7.58866  7.58867   10.00000 

40 

30 

7.00779  7.00779  10.00000 

30 

30 

7.59406  7.59406  10.00000 

30 

40 

7.02800  7.02800  10.00000 

20 

40 

7.59939  7.59939  10.00000 

20 

50 

7.04730  7.04730  10.00000 

10 

50 

7.60465   7.60466  10.00000 

10 

4   0 

7.06579  7.06579  10.00000 

0  56 

14    0 

7.60985   7.60986  10.00000 

046 

10 

7.08351   7.08352  10.00000 

50 

10 

7.61499  7.61500  10.00000 

50 

20 

7.10055   7.10055  10.00000 

40 

20 

7.62007  7.62008  10.00000 

40 

30 

7.11694  7.11694  10.00000 

30 

30 

7.62509  7.62510  10.00000 

30 

40 

7.13273  7.13273   10.00000 

20 

40 

7.63006  7.63006  10.00000 

20 

50 

7.14797  7.14797  10.00000 

10 

50 

7.63496  7.63497  10.00000 

10 

5  0 

7.16270  7.16270  10.00000 

0  55 

15    0 

7.63982  7.63982  10.00000 

045 

10 

7.17694  7.17694  10.00000 

50 

10 

7.64461   7.64462   10.00000 

50 

20 

7.19072  7.19073   10.00000 

40 

20 

7.64936  7.64937  10.00000 

40 

30 

7.20409  7.20409  10.00000 

30 

30 

7.65406  7.65406  10.00000 

30 

40 

7.21705  7.21705   10.00000 

20 

40 

7.65870  7.65871   10.00000 

20 

50 

7.22964  7.22964  10.00000 

10 

50 

7.66330  7.66330  10.00000 

10 

6  0 

7.24188  7.24188   10.00000 

0  54 

16    0 

7.66784  7.66785   10.00000 

044 

10 

7.25378  7.25378  10.00000 

50 

10 

7.67235   7.67235   10.00000 

50 

20 

7.26536  7.26536  10.00000 

40 

20 

7.67680  7.67680  10.00000 

40 

30 

7.27664  7.27664  10.00000 

30 

30 

7.68121   7.68121   10.00000 

30 

40 

7.28763  7.28764  10.00000 

20 

40 

7.68557  7.68558     9.99999 

20 

50 

7.29836  7.29836  10.00000 

10 

50 

7.68989  7.68990     9.99999 

10 

7   0 

7.30882  7.30882  10.00000 

0  53 

17    0 

7.69417  7.69418     9.99999 

043 

10 

7.31904  7.31904  10.00000 

50 

10 

7.69841   7.69842     9.99999 

50 

20 

7.32903  7.32903   10.00000 

40 

20 

7.70261   7.70261     9.99999 

40 

30 

7.33879  7.33879  10.00000 

30 

30 

7.70676  7.70677     9.99999 

30 

40 

7.34833  7.34833   10.00000 

20 

40 

7.71088  7.71088     9.99999 

20 

50 

7.35767  7.35767   10.00000 

10 

50 

7.71496  7.71496     9.99999 

10 

8   0 

7.36682  7.36682  10.00000 

0  52 

18    0 

7.71900  7.71900     9.99999 

042 

10 

7.37577  7.37577  10.00000 

50 

10 

7.72300  7.72301     9.99999 

50 

20 

7.38454  7.38455   10.00000 

40 

20 

7.72697  7.72697     9.99999 

40 

30 

7.39314  7.39315   10.00000 

30 

30 

7.73090  7.73090     9.99999 

30 

40 

7.40158  7.40158  10.00000 

20 

•    40 

7.73479  7.73480     9.99999 

20 

50 

7.40985   7.40985   10.00000 

10 

50 

7.73865   7.73866     9.99999 

10 

9  0 

7.41797  7.41797  1000000 

0  51 

19    0 

7.74248  7.74248     9.99999 

041 

10 

7.42594  7.42594  10.00000 

50 

10 

7.74627  7.74628     9.99999 

50 

20 

7.43376  7.43376  10.00000 

40 

20 

7.75003  7.75004     9.99999 

40 

30 

7.44145  7.44145   10.00000 

30 

30 

7.75376  7.75377     9.99999 

30 

40 

7.44900  7.44900  10.00000 

20 

40 

7.75745  7.75746     9.99999 

20 

50 

7.45643  7.45643  10.00000 

10 

50 

7.76112  7.76113     9.99999 

10 

10  0 

7.46373  7.46373  10.00000 

0  5O 

20    0 

7.76475  7.76476     9.99999 

04O 

/     // 

L  cos      L  cot        L  sin 

//      / 

/     // 

L  cos      L  cot        L  sin 

//    / 

89 


81 


/    // 

L  sin       L  tan       L  cos 

//     / 

/    // 

L  sin       L  tan       L  cos 

//    / 

2O  0 

7.76475  7.76476  9.99999 

0  4O 

3O    0 

7.94084  7.94086  9.99998 

03O 

10 

7.76836  7.76837  9.99999 

50 

10 

7.94325   7.94326  9.99998 

50 

20 

7.77  193  7.77  194  9.99  999 

40 

20 

7.94564  7.94566  9.99998 

40 

30 

7.77548  7.77549  9.99999 

30 

30 

7.94802  7.94804  9.99998 

30 

40 

7.77899  7.77900  9.99999 

20 

40 

7.95039  7.95040  9.99998 

20 

50 

7.78248  7.78249  9.99999 

10 

50 

7.95  274  7.95  276  9.99  998 

10 

21  0 

7.78594  7.78595  9.99999 

0  39 

31    0 

7.95508  7.95510  9.99998 

029 

10 

7.78938  7.78938  9.99999 

50 

10 

7.95  741   7.95  743  9.99  998 

50 

20 

7.79278  7.79279  9.99999 

40 

20 

7.95973   7.95974  9.99998 

40 

30 

7.79616  7.79617  9.99999 

30 

30 

7.96203   7.96205  9.99998 

30 

40 

7.79952  7.79952  9.99999 

20 

40 

7.96432   7.96434  9.99998 

20 

50 

7.80284  7.80285  9.99999 

10 

50 

/.96660  7.96662  9.99998 

10 

22  0 

7.80615  7.80615  9.99999 

0  38 

32    0 

7.96887  7.96889  9.99998 

028 

10 

7.80942  7.80943  9.99999 

50 

10 

7.97113   7.97114  9.99998 

50 

20 

7.81  268  7.81  269  9.99  999 

40 

20 

7.97337   7.97339  9.99998 

40 

30 

7.81591  7.81591  9.99999 

30 

30 

7.97560   7.97562  9.99998 

30 

40 

7.81911  7.81912  9.99999 

20 

40 

7.97782   7.97784  9.99998 

20 

50 

7.82229  7.82230  9.99999 

10 

50 

7.98003   7.98005  9.99998 

10 

23  0 

7.82545  7.82546  9.99999 

0  37 

33    0 

7.98223   7.98225  9.99998 

027 

10 

7.82859  7.82860  9.99999 

50 

10 

7.98442  7.98444  9.99998 

50 

20 

7.83170  7.83171  9.99999 

40 

20 

7.98660  7.98662  9.99998 

40 

30 

7.83479  7.83480  9.99999 

30 

30 

7.98876  7.98878  9.99998 

30 

40 

7.83786  7.83787  9.99999 

20 

40 

7.99092   7.99094  9.99998 

20 

50 

7.84091  7.84092  9.99999 

10 

50 

7.99306  7.99308  9.99998 

10 

24  0 

7.84393  .7.84394  9.99999 

0  36 

34    0 

7.99520  7.99522  9.99998 

026 

10 

7.84694  7.84695  9.99999 

50 

10 

7.99732  7.99734  9.99998 

50 

20 

7.84992  7.84993  9.99999 

40 

20 

7.99943   7.99946  9.99998 

40 

30 

7.85289  7.85290  9.99999 

30 

30 

8.00154  8.00156  9.99998 

30 

40 

7.85583  7.85584  9.99999 

20 

40 

8.00363  8.00365  9.99998 

20 

50 

7.85876  7.85877  9.99999 

10 

50 

8.00571  8.00574  9.99998 

10 

25  0 

7.86166  7.86167  9.99999 

0  35 

35    0 

8.00779  8.00781  9.99998 

025 

10 

7.86455  7.86456  9.99999 

50 

10 

8.00985   8.00987  9.99998 

50 

20 

7.86741  7.86743  9.99999 

40 

20 

8.01  190  8.01  193  9.99  998 

40 

30 

7.87026  7.87027  9.99999 

30 

30 

8.01395   8.01397  9.99998 

30 

40 

7.87309  787310  9.99999 

20 

40 

8.01  598  8.01  600  9.99  998 

20 

50 

7.87590  7.87591  9.99999 

10 

50 

8.01801  S.01'803  9.99998 

10 

26  0 

7.87870  7.87871  9.99999 

0  34 

36    0 

8.02002  8.02004  9.99998 

024 

10 

7.88147  7.88148  9.99999 

50 

10 

8.02203  8.02205  9.99998 

50 

20 

7.88423  7.88424  9.99999 

40 

20 

8.02402  8.02405  9.99998 

40 

30 

7.88697  7.88698  9.99999 

30 

30 

8.02601  8.02604  9.99998 

30 

40 

7.88969  7.88970  9.99999 

20 

40 

8.02799  8.02801  9.99998 

20 

50 

7.89240  7.89241  9.99999 

10 

50 

8.02996  8.02998  9.99998 

10 

27  0 

7.89509  7.89510  9.99999 

0  33 

37    0 

8.03  192  8.03  194  9.99  997 

023 

10 

7.89776  7.89777  9.99999 

50 

10 

8.03387  8.03390  9.99997 

50 

20 

7.90041   7.90043  9.99999 

40 

20 

8.03581   8.03584  9.99997 

40 

30 

7.90305  7.90307  9.99999 

30 

30 

8  03  775   8.03  777  9.99  997 

30 

40 

7.90568  7.90569  9.99999 

20 

40 

8.03967  8.03970  9.99997 

20 

50 

7.90829  7.90830  9.99999 

10 

50 

8.04  159  8.04  162  9.99  997 

10 

28  0 

7.91088  7.91089  9.99999 

0  32 

38    0 

8.04350  8.04353  9.99997 

022 

10 

7.91346  7.91347  9.99999 

50 

10 

804540  8.04543  9.99997 

50 

20 

7.91602  7.91603  9.99999 

40 

20 

8.04729  8.04732  9.99997 

40 

30 

7.91857  7.91858  9.99999 

30 

30 

8.04918  8.04921  9.99997 

30 

40 

7.92110  7.92111  9.99998 

20 

40 

8.05  105  8.05  108  9.99  997 

20 

50 

7.92362  7.92363  9.99998 

10 

50 

8.05292  8.05295  9.99997 

1(5 

29  0 

7.92612  7.92613  9.99998 

0  31 

39    0 

8.05  478  8.05  481  9.99  997 

021 

10 

7.92861   7.92862  9.99998 

50 

10 

8.05663  8.05666  9.99997 

50 

20 

7.93108  7.93110  9.99998 

40 

20 

8.05848  8.05851  9.99997 

40 

30 

7.93354  7.93356  9.99998 

30 

30 

8.06031  8.06034  9.99997 

30 

40 

7.93599  7.93601  9.99998 

20 

40 

8.06214  8.06217  9.99997 

20 

50 

7.93842  7.93844  9.99998 

10 

50 

8.06396  8.06399  9.99997 

10 

3O  0 

7.94084  7.94086  9.99998 

0  3O 

4O    0 

8.06578  8.06581  9.99997 

02O 

/    // 

L  cos       L  cot       L  sin 

//     / 

/     // 

L  cos       L  cot       L  sin 

//    / 

89 


/    // 

L  sin       L  tan      L  cos 

//      / 

/     // 

L  sin       L  tan      L  cos 

//    / 

4O  0 

8.06578  8.06581  9.99997 

0  2O 

5O    0 

8.16268  8.16273  9.99995 

01O 

10 

806758  8.06761  9.99997 

50 

10 

8.16413  8.16417  9.99995 

50 

20 

8.06938  8.06941  9.99997 

40 

20 

8.16557  8.16561   9.99995 

40 

30 

8.07117   8.07120  9.99997 

30 

30 

8.16700  8.16705   9.99995 

30 

40 

8.07295  8.07298  9.99997 

20 

40 

8.16843  8.168-18  9.99995 

20 

50 

8.07473   8.07476  9.99997 

10 

50 

8.16986  8.16991   9.99995 

10 

41    0 

8.07650  8.07653  9.99997 

0  19 

51     0 

8.17128  8.17133   9.99995 

0    9 

10 

8.07826  8.07829  9.99997 

50 

10 

8.17270  8.17275   9.99995 

50 

20 

8.08002  8.08005  9.99997 

40 

20 

8.17411  8.17416  9.99995 

40 

30 

8.08176  8.08180  9.99997 

30 

30 

8.17552  8.17557  9.99995 

30 

40 

8.08350  8.08354  9.99997 

20 

40 

8.17692  8.17697  9.99995 

20 

50 

8.08524  8.08527  9.99997 

10 

50 

8.17832  8.17837  9.99995 

10 

42   0 

8.08696  8.08700  9.99997 

0  18 

52    0 

8.17971  8.17976  9.99995 

0    8 

10 

8.08868  8.08872  9.99997 

50 

10 

8.18110  8.18115   9.96995 

50 

20 

8.09040  8.09043  9.99997 

40 

20 

8.18249  8.18254  9.99995 

40 

30 

8.09210  8.09214  9.99997 

30 

30 

8.18387  8.18392  9.99995 

30 

40 

8.09  380  8.09384  9.99997 

20 

40 

8.18524  8.18530  999995 

20 

50 

8.09550  8.09553  9.99997 

10 

50 

8.18662  8.18667  9.99995 

10 

43   0 

8.09718  8.09722  9.99997 

0  17 

53    0 

8.18798  8.18804  9.99995 

0     7 

10 

8.09886  8.09890  9.99997 

50 

10 

8.18935  8.18940  9.99995 

50 

20 

8.10054  8.10057  9.99997 

40 

20 

8.19071  8.19076  9.99995 

40 

30 

8.10220  8.10224  9.99997 

30 

30 

8.19206  8.19211   9.99995 

30 

40 

8.10386  8.10390  9.99996 

20 

40 

8.19341   8.19347  9.99995 

20 

50 

8.10552  8.10555  9.99996 

10 

50 

8.19476  8.19481   9.99995 

10 

44   0 

8.10717  8.10720  9.99996 

0  16 

54    0 

8.19610  8.19616  9.99995 

0    6 

10 

8.10881  8.10884  9.99996 

50 

10 

8.19744  8.19749  9.99995 

50 

20 

8.11044  8.11018  9.99996 

40 

20 

8.19877  8.19883   9,99995 

40 

30 

811207   8.11  211  9.99996 

30 

30 

8.20010  8.20016  9-99995 

30 

40 

8.11370  8.11373  9.99996 

20 

40 

8.20143  8.20149  9.99995 

20 

50 

8.11531  8.11535  9.99996 

10 

50 

8.20275   8.20281   9.99994 

10 

45    0 

8.11693  8.11696  9.99996 

0  15 

55    0 

8.20407  8.20413  9.99994 

0    5 

10 

8.11853  8.11857  9.99996 

50 

10 

8.20  538  8.20  544  9.99  994 

50 

20 

8.12013  8.12017  9.99996 

40 

20 

8.20669  820675   9.99994 

40 

30 

8.12172  8.12  176  9.99996 

30 

30 

8.20800  8.20806  9.99994 

30 

40 

8.12331   812335  9.99996 

20 

40 

8.20930  8.20936  9.99994 

20 

50 

8.12489  8.12*493  9.99996 

10 

50 

8.21060  8.21066  9.99994 

10 

46  0 

8.12647  8.12651  9.99996 

0  14 

56    0 

8.21189  8.21  195  9.99994 

0    4 

10 

8.12804  8.12808  9.99996 

50 

10 

8.21319  8.21324  9.09994 

50 

20 

8.12961   8.12965  9.99996 

40 

20 

8.21  447   8.21  453  9.99  994 

40 

30 

8.13117  8.13  121  9.99996 

30 

30 

8.21  576  8.21581  9.99994 

30 

40 

8.13272  8.13276  9.99996 

20 

40 

8.21  703  8.21  709  9.99  994 

20 

50 

8.13427  8.13431  9.99996 

10 

50 

8.21831   8.21  837  9.99994 

10 

47   0 

8.13581   8.13585   9.99996 

0  13 

57    0 

8.21958  8.21964  9.99994 

0    3 

10 

8.13735  8.13  739  9.99996 

50 

10 

8.22085   8.22091   9.99994 

50 

20 

8.13888  8.13892  9.99996 

40 

20 

8.22211   8.22217  9.99994 

40 

30 

8.14041   8.14045  9.99996 

30 

30 

822337  8.22343  9.99994 

30 

40 

8.14193  8.14197  9.99996 

20 

40 

822463  8.22-169  9.99994 

20 

50 

8.14344  8.143-18  9.99996 

10 

50 

8.22588  8.22595  9.99994 

10 

48   0 

8.14495  8.14500  9.99996 

0  12 

58    0 

8.22713  8.22720  9.99994 

0    2 

10 

8.14646  8.14650  9.99996 

50 

10 

8.22838  8.22844  9.99994 

50 

20 

8.14796  8.14800  9.99996 

40 

20 

8.22962   8.22968  9.99994 

40 

30 

814945  8.14950  9.99996 

30 

30 

8.23086  8.23092  9.99994 

30 

40 

8.15094  8.15099  9.99996 

20 

40 

8.23210  8.23216  9.99994 

20 

50 

8.15243  8.15247  9.99996 

10 

50 

8.23333  8.23339  9.99994 

10 

49   0 

8.15391  8.15395  9.99996 

0  11 

59    0 

8.23456  8.23462  999994 

0     1 

10 

8.15538  8.15543  9.99996 

50 

10 

8.23578  8.23585   9.99994 

50 

20 

8.15685   8.15690  9.99996 

40 

20 

8.23  700  8.23  707  9.99  994 

40 

30 

8.15832  8.15836  9.99995 

30 

30 

8.23822  8.23829  9.99993 

30 

40 

8.15978  8.15982  9.99995 

20 

40 

8.23944  8.23950  9.99993 

20 

50 

8.16123  8.16128  9.99995 

10 

50 

8.24065  8.24071  9.99993 

10 

50  0 

8.16268  8.16273  9.99995 

0  1C 

6O    0 

8.24186  8.24192  9.99993 

0    O 

/     // 

L  cos      L  cot       L  sin 

//      / 

/     // 

L  cos      L  cot       L  sin 

//    / 

89' 


33 


/  // 

L  sin       L  tan       L  cos 

//     / 

/     // 

L  sin       L  tan       L  cos 

//    / 

O   0 

8.24  186  8.24  192  9.99  993 

0  6O 

1O    0 

8.30879  8.30888  9.99991 

05O 

10 

8.24306  8.24313  9.99993 

50 

10 

8.30983  8.30992  9.99991 

50 

20 

8.24426  8.24433  9.99993 

40 

20 

8.31086  8.31095  9.99991 

40 

30 

824546  8.24553  9.99993 

30 

30 

8.31188  8.31198  9.99991 

30 

40 

8.24665   824672  9.99993 

20 

40 

8.31291   8.31300  9.99991 

20 

50 

8.24785   8.24791  9.99993 

10 

50 

8.31393  8.31403  9.99991 

10 

1    0 

8.24903  8.24910  9.99993 

0  59 

11     0 

8.31495  8.31505  9.99991 

049 

10 

8.25022  8.25029  9.99993 

50 

10 

8.31597  8.31606  9.99991 

50 

20 

8.25  140  8.25  147  9.99  993 

40 

20 

8.31699  8.31  708  9.99991 

40 

30 

8.25258  8.25265   9.99993 

30 

30 

8.31800  8.31809  9.99991 

30 

40 

.8.25375  8.25382  9.99993 

20 

40 

8.31901  8.31911   9.99991 

20 

50 

8.25493  8.25500  9.99993 

10 

50 

8.32002  8.32012   9.99991 

10 

2    0 

8.25609  8.25616  9.99993 

0  58 

12    0 

8.32103  8.32112  9.99990 

048 

10 

8.25  726  8.25  733  9.99  993 

50 

10 

8.32203  8.32213  9.99990 

50 

20 

8.25842  8.25849  9.99993 

40 

20 

8.32303  8.32313  9.99990 

40 

30 

8.25958  8.25965   9.99993 

30 

30 

8.32403  8.32413  9.99990 

30 

40 

8.26074  8.26081   9.99993 

20 

40 

8.32503  8.32513  9.99990 

20 

50 

8.26189  8.26196  9.99993 

10 

50 

8.32602  8.32612  9.99990 

10 

3  0 

8.26304  8.26312  9.99993 

0  57 

13    0 

8.32702  8.32711  9.99990 

047 

10 

8.26419  826426  9.99993 

50 

10 

8.32801  8.32811  9.99990 

50 

20 

8.26553  8.26541  9.99993 

40 

20 

8.32899  8.32909  9.99990 

40 

30 

8.26648  8.26655  9.99993 

30 

30 

8.32998  8.33008  9.99990 

30 

40 

8.26761   8.26769  9.99993 

20 

40 

8.33096  8.33106  9.99990 

20 

50 

8.26875  8.26882  9.99993 

10 

50 

8.33195  8.33205  9.99990 

10 

4   0 

8.26988  8.26996  9.99992 

0  56 

14    0 

8.33292  8.33302  9.99990 

046 

10 

8.27101  8.27109  9.99992 

50 

10 

8.33390  8.33400  9.99990 

50 

20 

8.27214  8.27221   9.99992 

40 

20 

8.33488  8.33498  9.99990 

40 

30 

8.27326  8.27334  9.99992 

30 

30 

8.33585  8.33595  9.99990 

30 

40 

8.27438  8.27446  9.99992 

20 

40 

8.33682  8.33692  9.99990 

20 

50 

8.27550  8.27558  9.99992 

10 

50 

8.33779  8.33789  9.99990 

10 

5   0 

8.27661  8.27669  9.99992 

0  55 

15    0 

8.33875  8.33886  9.99990 

045 

10 

8.27773  8.27780  9.99992 

50 

10 

8.33972  8.33982  9.99990 

50 

20 

8.27883  8.27891  9.99992 

40 

20 

8.34086  8.34078  9.99990 

40 

30 

8.27994  8.28002  9.99992 

30 

30 

8.34164  8.34174  9.99990 

30 

40 

8.28104  828112  9.99992 

20 

40 

8.34260  8.34270  9.99989 

20 

50 

8.28215  8.28223  9.99992 

10 

50 

8.34355  8.34366  9.99989 

10 

6   0 

8.28324  8.28332  9.99992 

0  54 

16    0 

8.34450  8.34461  9.99989 

044 

10 

8.28434  8.28442  9.99992 

50 

10 

8.34546  8.34556  9.99989 

50 

20 

8.28543  8.28551  9.99992 

40 

20 

8.34640  8.34651  9.99989 

40 

30 

8.28652  8.28660  9.99992 

30 

30 

8.34735  8.34746  9.99989 

30 

40 

8.28761  8.28769  9.99992 

20 

40 

8.34830  8.34840  9.99989 

20 

50 

8.2S869  8.28877  9.99992 

10 

50 

8.34924  8.34935   9.99989 

10 

7    0 

8.28977  8.28986  9.99992 

0  53 

17    0 

8.35  018  8.35  029  9.99  989 

043 

10 

8.29085  8.29094  9.99992 

50 

10 

8.35112  8.35  123  9.99989 

50 

20 

8.29093  8.29201   9.99992 

40 

20 

8.35206  8.35217  9.99989 

40 

30 

8.29300  8.29309  9.99992 

30 

30 

8.35  299  8.35  310  9.99  989 

30 

40 

8.29407  8.29416  9.99992 

20 

40 

8.35  392  8.35  403  9.99  989 

20 

50 

8.29514  8.29523  9.99992 

10 

50 

8.35  485  8.35  497  9.99  989 

10 

8  0 

8.29621  829629  9.99992 

0  52 

18    0 

8.35578  8.35590  9.99989 

042 

10 

8.29727  8.29736  9.99991 

50 

10 

8.35671   8.35682  9.99989 

50 

20 

8.29833  8.29842  9.99991 

40 

20 

8.35764  8.35  775  9.99989 

40 

30 

8.29939  8.29947  9.99991 

30 

30 

8.35  856  8.35  867  9.99  989 

30 

40 

8.30044  8.30053   9.99991 

20 

40 

8.35948  8.35959  9.99989 

20 

-50 

8.30150  8.30158  9.99991 

10 

50 

8.36040  8.36051  9.99989 

10 

9   0 

8.30255  8.30263  9.99991 

0  51 

19    0 

8.36131  8.36143  9.99989 

041 

10 

8.30359  8.30368  9.99991 

50 

10 

8.36223  8.36235   9.99988 

50 

20 

8.30464  8.30473  9.99991 

40 

20 

8.36314  8.36326  9.99988 

40 

30 

8.30568  8.30577  9.99991 

30 

30 

8.36405  8.36417  9.99988 

30 

40 

8.30672  8.30681  9.99991 

20 

40 

836496  8.36508  9.99988 

20 

50 

8.30776  8.30785  9.99991 

50 

8.36587  8.36599  9.99988 

10 

10   0 

8.30879  8.30888  9.99991 

*0  5O 

2O    0 

8.36678  8.36689  9.99988 

04O 

/    // 

L  cos        L  cot       L  sin 

//     / 

/     // 

L  cos        L  cot       L  sin 

//    / 

88C 


34 


/    // 

L  sin      L  tan      L  cos 

//      / 

/    // 

L  sin      L  tan      L  cos 

//    / 

2O  0 

8.36678  8.36689  9.99988 

0  40 

3O    0 

8.41792  8.41807  9.99985 

03O 

10 

8.36768  8.36780  9.99  988 

50 

10 

8.41872  8.41887  9.99985 

50 

20 

8.36858  8.36870  9.99988 

40 

20 

8.41952  8.41967  9.99985 

40 

30 

8.36948  8.36960  9.99988 

30 

30 

8.42032  8.42048  9.99985 

30 

40 

8.37038  8.37050  9.99988 

20 

40 

8.42112  8.42127  9.99985 

20 

50 

8.37128  8.37140  9.99988 

10 

50 

8.42192  8.42207  9.99985 

10 

21    0 

8.37217  8.37229  9.99988 

0  39 

31    0 

8.42272  8.42287  9.99985 

029 

10 

8.37306  8.37318  9.99988 

50 

10 

8.42351  8.42366  9.99985 

50 

20 

8.37395  8.37408  9.99988 

40 

20 

8.42430  8.42446  9.99985 

40 

30 

8.37484  8.37497  9.99988 

30 

30 

8.42510  8.42525  9.99985 

30 

40 

8.37573  8.37585  9.99988 

20 

40 

8.42589  8.42406  9.99985. 

20 

50 

8.37662  8.37674  9.99988 

10 

50 

8.42667  8.42683  9.99985 

10 

22   0 

8.37750  8.37762  9.99988 

0  38 

32    0 

8.42746  8.42762  9.99984 

028 

10 

8.37838  8.37850  9.99988 

50 

10 

8.42825  8.42840  9.96984 

50 

20 

8.37926  8.37938  9.99988 

40 

20 

8.42903  8.42919  9.99984 

40 

30 

8.38014  8.38026  9.99987 

30 

30 

8.42  982  8.42997  9.99984 

30 

40 

8.38101  8.38114  9.99987 

20 

40 

8.43060  8.43075  9.99984 

20 

50 

8.38189  8.33202  9.99987 

10 

50 

8.43  138  8.43  154  9.99  984 

10 

23   0 

8.38276  8.38289  9.99987 

0  37 

33    0 

8.43  216  8.43  232  9.99  984 

027 

10 

8.38363  8.38376  9.99987 

50 

10 

8.43293  8.43309  9.99984 

50 

20 

8.38450  8.38463  9.99987 

40 

20 

8.43371  8.43387  9.99984 

40 

30 

8.38537  8.38550  9.99987 

30 

30 

8.43448  8.43464  9.99984 

30 

40 

8.38624  838636  9.99987 

20 

40 

8.43526  8.43542  9.99984 

20 

50 

8.38710  8.38723  9.99987 

10 

50 

8.43603  8.43619  9.99984 

10 

24  0 

8.38796  8.38809  9.99987 

0  36 

34    0 

8.43680  8.43696  9.99984 

026 

10 

8.38882  8.38895  9.99987 

50 

10 

8.43757  8.43773  9.99984 

50 

20 

8.38968  8.38981  9.99987 

40 

20 

8.43834  8.43850  9.99984 

40 

30 

8.39054  8.39067  9.99987 

30 

30 

8.43910  8.43927  9.99984 

30 

40 

8.39139  8.39153  9.99987 

20 

40 

8.43987  8.44003  9.99984 

20 

50 

8.39225  8.39238  9.99987 

10 

50 

8.44063  8.44080  9.99983 

10 

25  0 

8.39310  8.39323  9.99987 

0  35 

35    0 

8.44139  8.44156  9.99983 

025 

10 

8.39395  8.39408  9.99987 

50 

10 

8.44216  8.44232  9.99983 

50 

20 

8.39480  8.39493  9.99987 

40 

20 

8.44292  8.44308  9.99983 

40 

30 

8.39565  8.39587  9.99987 

30 

30 

8.44367  8.44384  9.99983 

30 

40 

8.39649  8.39663  9.99987 

20 

40 

8.44443  8.44460  9.99983 

20 

50 

8.39734  8.39747  9.99986 

10 

50 

8.44519  8.44536  9.99983 

10 

26   0 

8.39818  8.39832  9.99986 

0  34 

36    0 

8.44594  8.44611  9.99983 

024 

10 

8.39902  8.39916  9.99986 

50 

10 

8.44669  8.44686  9.99  983 

50 

20 

8.39986  8.40000  9.99986 

40 

20 

8.44745  8.44762  9.99983 

40 

30 

8.40070  8.40083  9.99986 

30 

30 

8.44820  8.44837  9.99983 

30 

40 

8.40153  8.40167  9.99986 

20 

40 

8.44895  8.44912  9.99983 

20 

50 

8.40237  8.40251  9.99986 

10 

50 

8.44969  8.44987  9.99983 

10 

27   0 

8.40320  8.40334  9.99986 

0  33 

37    0 

8.45044  8.45061  9.99983 

023 

10 

8.40403  8.40417  9.99986 

50 

10 

8.45119  8.45  136  9.99983 

50 

20 

8.40486  8.40500  9.99986 

40 

20 

8.45  193  8.45  210  9.99  983 

40 

30 

8.40569  8.40583  9.99986 

30 

30 

8.45267  8.45285  9.99983 

30 

40 

8.40651  8.40665  9.99986 

20 

40 

8.45341  8.45359  9.99982 

20 

50 

8.40734  8.40748  9.99986 

10 

50 

8.45415  8.45433  9.99982 

10 

28   0 

8.40816  8.40830  9.99986 

0  32 

38    0 

8.45489  8.45507  9.99982 

022 

10 

8.40898  8.40913  9.99986 

50 

10 

8.45563  8.45581  9.99982 

50 

20 

8.40  980  8.40995  9.99986 

40 

20 

8.45637  8.45655  9.99982 

40 

30 

8.41062  8.41077  9.99986 

30 

30 

8.45  710  8.45  728  9.99  982 

30 

40 

841144  8.41158  9.99986 

20 

40 

8.45  784  8.45  802  9.99  982 

20 

50 

8.41225  8.41240  9.99986 

10 

50 

8.45857  8.45875  9.99982 

10 

29   0 

8.41307  8.41321  9.99985 

0  31 

39    0 

8.45930  8.45948  9.99982 

021 

10 

8.41388  8.41403  9.99985 

50 

10 

8.46003  8.46021  9.99982 

50 

20 

8.41469  8.41484  9.99985 

40 

20 

8.46076  8.46094  9.99982 

40 

30 

8.41550  8.41565  9.99985 

30 

30 

8.46149  8.46167  9.99982 

30 

40 

8.41631  8.41646  9.99985 

20 

40 

8.46222  8.46240  9.99982 

20 

50 

8.41711  8.41726  9.99985 

10 

50 

8.46294  8.46312  9.99982 

10 

3O  0 

8.41  792  8.41  807  9.99  985 

0  30 

4O    0 

8.46366  8.46385  9.99932 

020 

/     // 

L  cos      L  cot       L  sin 

//      / 

/     // 

L  cos       L  cot       L  sin 

//    / 

1  i 

88' 


/    // 

L  sin       L  tan       L  cos 

//     / 

/     // 

L  sin       L  tan       L  cos 

//    / 

40   0 

8.46366  8.46385  9.99982 

0  2O 

5O    0 

8.50504  8.50527  9.99978 

010 

10 

8.46439  8.46457  9.99982 

50 

10 

8.50570  8.50593  9.99978 

50 

20 

8.46511  8.46529  9.99982 

40 

20 

8.50636  8.50658  9.99978 

40 

30 

8.46583  8.46602  9.99981 

30 

30 

8.50701  8.50724  9.99978 

30 

40 

8.46655    8.46674  9.99981 

20 

40 

8.50767  8.50789  9.99977 

20 

50 

8.46727  8.46745  9.99981 

10 

50 

8.50832  8.50855  9.99977 

10 

41    0 

8.46799  8.46817  9.99981 

0  19 

51     0 

8.50897  8.50920  9.99977 

0    9 

>        10 

8.46870  8.46889  9.99981 

50 

10 

8.50963  8.50985  9.99977 

50 

20 

8.46942  8.46960  9.99981 

40 

20 

8.51028  8.51050  9.99977 

40 

30 

8.47013  8.47032  9.99981 

30 

30 

8.51092  8.51015  9.99977 

30 

40 

8.47084  8.47103  9.99981 

20 

40 

8.51157  8.51180  9-99977 

20 

50 

8.47155  8.47174  9.99981 

10 

50 

8.51222  8.51245  9.99977 

10 

42   0 

8.47226  8.47245  9.99981 

0  18 

52    0 

8.51287  8.51310  9.99977 

0    8 

10 

8.47297  8.47316  9.99981 

50 

10 

8.51351  8.51374  9.99977 

50 

20 

8.47368  847387  9.99981 

40 

20 

8.51416  8.51439  9.99977 

40 

30 

8.47439  8.47458  9.99981 

30 

30 

8.51480  851  503  9.99977 

30 

40 

8.47509  8.47528  9.99981 

20 

40 

8.51544  8.51568  9.99977 

20 

50 

8.47580  8.47599  9.99981 

10 

50 

8.51609  8.51632  9.99-977 

10 

43   0 

8.47650  8.47669  9.99981 

0  17 

53    0 

8.51673  8.51696  9.99977 

0     7 

10 

8.47720  8.47740  9.99980 

50 

10 

8.51737  8.51760  9.99976 

50 

20 

8.47790  8.47810  9.99980 

40 

20 

8.51801   8.51824  9.99976 

40 

30 

8.47860  8.47880  9.99980 

30 

30 

8.51864  8.51888  9.99976 

30 

40 

8.47930  8.47950  9.99980 

20 

40 

8.51928  8.51952  9.99976 

20 

50 

8.48000  8.48020  9.99980 

10 

50 

8.51992  8.52015  9.99976 

10 

44   0 

8.48096  8.48090  9.99980 

0  16 

54    0 

8.52055  8.52079  9.99976 

0    6 

10 

8.48139  8.48159  9.99980 

50 

10 

8.52119  8.52143  9.99976 

50 

20 

8.48208  8.48228  9.99980 

40 

20 

8.52  182  8.52  206  9.99  976 

40 

30 

8.48278  8.48298  9.99980 

30 

30 

8.52245  8.52269  9.99976 

30 

40 

8.48347  8.48367  9.99980 

20 

40 

8.52308  8.52332  9.99976 

20 

50 

8.48416  8.48436  9.99980 

10 

50 

8.52371  8.52396  9.99976 

10 

45   0 

8.48485  8.48505  9.99980 

0  15 

55    0 

8.52434  8.52459  9.99976 

0    5 

10 

8.48554  8.48574  9.99980 

50 

10 

8.52497  8.52522  9.99976 

50 

20 

8.48622  8.48643  9.99980 

40 

20 

8.52560  8.52584  9.99976 

40 

30 

8.48691  8.48711  9.99980 

30 

30 

8.52623  8.52647  9.99975 

30 

40 

8.48760  8.48780  9.99979 

20 

40 

8.52685  8.52710  9.99975 

20 

50 

8.48828  8.48849  9.99979 

10 

50 

8.52748  8.52772  9.99975 

10 

46   0 

8.48896  8.48917  9.99979 

0  14 

56    0 

8.52810  8.52835  9.99975 

0    4 

10 

8.48965  8.48985  9.99979 

50 

10 

8.52872  8.52897  9.99975 

50 

20 

8.49033  8.49053  9.99979 

40 

20 

8.52935  8.52960  9.99975 

40 

30 

8.49101  8.49121   9.99979 

30 

30 

8.52997  8.53022  9.99975 

30 

40 

8.49169  8.49189  9.99979 

20 

40 

8.53059  8.53084  9.99975 

20 

50 

8.49236  8.49257  9.99979 

10 

50 

8.53121  8.53146  9.99975 

10 

47   0 

8.49304  8.49325  9.99979 

0  13 

57    0 

8.53183  8.53208  9.99975 

0    3 

10 

8.49372  8.49393  9.99979 

50 

10 

8.53245  8.53270  9.99975 

50 

20 

8.49439  8.49460  9.99979 

40 

20 

8.53306  8.53332  9.99975 

40 

30 

8.49506  8.49528  9.99979 

30 

30 

8.53368  8.53393  9.99975 

30 

40 

8.49574  8.49595  9.99979 

20 

40 

8.53  429  8.53  455   9.99  975 

20 

50 

8.49641  8.49662  9.99979 

10 

50 

8.53491  8.53516  9.99974 

10 

48   0 

8.49708  8.49729  9.99979 

0  12 

58    0 

8.53552  8.53578  9.99974 

0    2 

10 

8.49775  8.49796  9.99979 

50 

10 

8.53614  8.53639  9.99974 

50 

20 

8.49842  8.49863  9.99978 

40 

20 

8.53675  8.53700  9.99974 

40 

30 

8.49908  8.49930  9.99978 

30 

30 

8.53736  8.53762  9.99974 

30 

40 

8.49975  8.49997  9.99978 

20 

40 

8.53797  8.53823  9.99974 

20 

50 

8.50042  8.50063  9.99978 

10 

50 

8.53858  8.53884  9.99974 

10 

49   0 

8.50108  850130  9.99978 

0  11 

59    0 

8.53919  8.53945  9.99974 

0     1 

10 

8.50174  8.50196  9.99978 

50 

10 

8.53979  8.54005  9.99974 

50 

20 

8.50241  8.50263  9.99978 

40 

20 

8.54040  8.54066  9.99974 

40 

30 

850307  8.50329  9.99978 

30 

30 

8.54101  8.54127  9.99974 

30 

40 

8.50373  8.50395  9.99978 

20 

40 

8.54161  8.54187  9.99974 

20 

50 

8.50439  8.50461  9.99978 

10 

50 

8.54222  8.54248  9.99974 

10 

5O  0 

8.50504  8.50527  9.99978 

0  1O 

6O    0 

8.54282  8.54308  9.99974 

0    O 

/    // 

L  cos       L  cot       L  sin 

//     / 

/     // 

L  cos       L  cot       L  sin 

//    / 

36 


/ 

SLsin  SLtan  llLcot  9Lcos 

/ 

o 

.24186  .24192  .75808  .99993 

60 

1 

.24903  .24910  .75090  .99993 

59 

2 

.25609  .25616  .74384  .99993 

58 

3 

.26304  .26312  .73688  .99993 

57 

4 

.26988  .26996  .73004  .99992 

56 

5 

.27661  .27669  .72331  .99992 

55 

6 

.28324  .28332  .71668  .99992 

54 

7 

.28977  .28986  .71014  .99992 

53 

8 

.29621  .29629  .70371  .99992 

52 

9 

.30255  .30263  .69737  .99991 

51 

1C 

.30879  .30888  .69112  .99991 

50 

11 

.31495  .31505  .68495   .99991 

49 

12 

.32103  .32112  .67888  .99990 

48 

13 

.32702  .32711  .67289  .99990 

47 

14 

.33292  .33302  .66698  .99990 

46 

15 

.33875  .33886  .66114  .99990 

45 

16 

.34450  .34461  .65539  .99989 

44 

17 

.35018  .35029  .64971  .99989 

43 

18 

.35578  .35590  .64410  .99989 

42 

19 

.36131  .36143  .63857  .99989 

41 

20 

.36678  .36689  .63311  .99988 

40 

21 

.37217  .37229  .62771   .99988 

39 

22 

.37750  .37762  .62238  .99988 

38 

23 

.38276  .38289  .61711  .99987 

37 

24 

.38796  .38809  .61191  .99987 

36 

25 

.39310  .39323  .60677  .99987 

35 

26 

.39818  .39832  .60168  .99986 

34 

27 

.40320  .40334  .59666  .99986 

33 

28 

.40816  .40830  .59170  .99986 

32 

29 

.41307  .41321  .58679  .99985 

31 

3O 

.41792  .41807  .58193  .99985 

30 

31 

.42272  .42287  .57713  .99985 

29 

32 

.42746  .42762  .57238  .99984 

28 

33 

.43216  .43232  .56768  .99984 

27 

34 

.43680  .43696  .56304  .99984 

26 

35 

.44139  .44156  .55844  .99983 

25 

36 

.44594  .44611  .55389  .99983 

24 

37 

.45044  .45061  .54939  .99983 

23 

38 

.45489  .45507  .54493  .99982 

22 

39 

.45930  .45948  .54052  ,99982 

21 

40 

.46366  .46385  .53615  .99982 

2O 

41 

.46799  .46817  .53183  .99981 

19 

42 

.47226  .47245  .52755  .99981 

18 

43 

.47650  .47669  .52331  .99981 

17 

44 

.48069  .48089  .51911  .99980 

16 

45 

.48485  .48505  .51495  .99980 

15 

46 

.48896  .48917  .51083  .99979 

14 

47 

.49304  .49325  .50675  .99979 

13 

48 

.49708  .49729  .50271  .99979 

12 

49 

.50108  .50130  .49870  .99978 

11 

50 

.50504  .50527  .49473  .99978 

1O 

51 

.50897  .50920  .49080  .99977 

9 

52 

.51287  .51310  .48690  .99977 

8 

53 

.51673  .51696  .48304  .99977 

7 

54 

.52055  .52079  .47921  .99976 

6 

55 

.52434  .52459  .47541  .99976 

5 

56 

.52810  .52835  .47165  .99975 

4 

57 

.53183  .53208  .46792  .99975 

3 

58 

.53552  .53578  .46422  .99974 

2 

59 

.53919  .53945  .46055  .99974 

1 

60 

.54282  .54308  .45692  .99974 

O 

/ 

SLcos  SLcot  ULtan  9Lsin 

/ 

/ 

SLsin  SLtan  11  L  cot  9Lcos 

/ 

O 

.54282  .54308  .45692  .99974 

60 

1 

.54642  .54669  .45331  .99973 

59 

2 

.54999  .55027  .44973  .99973 

58 

3 

.55354  .55382  .44618  .99972 

57 

4 

.55705  .55734  .44266  .99972 

56 

5 

.56054  .56083  .43917  .99971 

55 

6 

.56400  .56429  .43571  .99971 

54 

7 

.56743  .56773   .43227  .99970 

53 

8 

.57084  .57114  .42886  .99970 

52 

9 

.57421  .57452  .42548  .99969 

51 

10 

.57757  .57788  .42212  .99969 

r>o 

11 

.58089  .58121  .41879  .99968 

49 

12 

.58419  .58451   .41549  .99968 

48 

13 

.58747  .58779  .41221  .99967 

47 

14 

.59072  .59105  .40895  .99967 

46 

15 

.59395  .59428  .40572  .99967 

45 

16 

.59715  .59749  .40251  .99966 

44 

17 

.60033  .60068  .39932  .99966 

43 

18 

.60349  .60384  .39616  .99965 

42 

19 

.60662  .60698  .39302  .99964 

41 

20 

.60973   .61009  .38991  .99964 

4O 

21 

.61282  .61319  .38681  .99963 

39 

22 

.61589  .61626  .38374  .99963 

38 

23 

.61894  .61931  .38069  .99962 

37 

24 

.62196  .62234  .37766  .99962 

36 

25 

.62497  .62535  .37465  .99961 

35 

26 

.62795  .62834  .37166  .99961 

34 

27 

.63091  .63131  .36869  .99960 

33 

28 

.63385  .63426  .36574  .99960 

32 

29 

.63678  .63718  .36282  .99959 

31 

30 

.63968  .64009  .35991  .99959 

30 

31 

.64256  .64298  .35702  .99958 

29 

32 

.64543  .64585  .35415  .99958 

28 

33 

.64827  .64870  .35130  .99957 

27 

34 

.65110  .65154  .34846  .99956 

26 

35 

.65391  .65435  .34565  .99956 

25 

36 

.65670  .65715  .34285  .99955 

24 

37 

.65947  .65993  .34007  .99955 

23 

38 

.66223  .66269  .33731  .99954 

22 

39 

.66497  .66543  .33457  .99954 

21 

4O 

.66769  .66816  .33184  .99953 

2O 

41 

.67039  .67087  .32913  .99952 

19 

42 

.67308  .67356  .32644  .99952 

18 

43 

.67575   .67624  .32376  .99951 

17 

44 

.67841  .67890  .32110  .99951 

16 

45 

.68104  .68154  .31846  .99950 

15 

46 

.68367  .68417  .31583  .99949 

14 

47 

.68627  .68678  .31322  .99949 

13 

48 

.68886  .68938  .31062  .99948 

12 

49 

.69144  .69196  .30804  .99948 

11 

50 

.69400  .69453  .30547  .99947 

10 

51 

.69654  .69708  .30292  .99946 

9 

52 

.69907  .69962  .30038  .99946 

8 

53 

.70159  .70214  .29786  .99945 

7 

54 

.70409  .70465  .29535  .99944 

6 

55 

.70658  .70714  .29286  .99944 

5 

56 

.70905  .70962  .29038  .99943 

4 

57 

.71151   .71208  .28792  .99942 

3 

58 

.71395  .71453  .28547  .99942 

2 

59 

.71638  .71697  .28303  .99941 

1 

60 

.71880  .71940  .28060  .99940 

O 

/ 

SLcos  SLcot  ULtan  9Lsin 

/ 

88C 


87' 


37 


/ 

SLsin  SLtan  llLcot  9Lcos 

/ 

o 

.71880  .71940  .28060  .99940 

60 

1 

.72120  .72181   .27819  .99940 

59 

2 

.72359  .72420  .27580  .99939 

58 

3 

.72597  .72659  .27341  .99938 

57 

4 

.72834  .72896  .27104  .99938 

56 

5 

.73069  .73132  .26868  .99937 

55 

*     6 

.73303  .73366  .26634  .99936 

54 

7 

.73535   .73600  .26400  .99936 

53 

8 

.73767  .73832   .26168  .99935 

52 

9 

.73997  .74063   .25937  .99934 

51 

10 

.74226  .74292  .25708  .99934 

50 

11 

.74454  .74521  .25479  .99933 

49 

12 

.74680  .74748  .25252   .99932 

48 

13 

.74906  .74974  .25026  .99932 

47 

14 

.75130  .75199  .24801   .99931 

46 

IS 

.75353  .75423  .24577  .99930 

45 

16 

.75575   .75645   .24355   .99929 

44 

17 

.75795  .75867  .24133   .99929 

43 

18 

.76015   .76087   .23913  .99928 

42 

19 

.76234  .76306  .23694  .99927 

41 

20 

.76451   .76525   .23475   .99926 

4O 

21 

.76667  .76742  .23258  .99926 

39 

22 

.76883  .76958  .23042  .99925 

38 

23 

.77097  .77173   .22827  .99924 

37 

24 

.77310  .77387   .22613  .99923 

36 

25 

.77522  .77600  .22400  .99923 

35 

26 

.77733  .77811   .22189  .99922 

34 

27 

.77943  .78022   .21978  .99921 

33 

28 

.78152  .78232  .21768  .99920 

32 

29 

.78360  .78441   .21559  .99920 

31 

30 

.78568  .78649  .21351   .99919 

30 

31 

.78774  .78855   .21145   .99918 

29 

32 

.78979  .79061   .20939.  .99917 

28 

33 

.79183   .79266  .20734  .99917 

27 

34 

.79386  .79470  .20*530  .99916 

26 

35 

.79588  .79673   .20327  .99915 

25 

36 

.79789  .79875   .20125  .99914 

24 

37 

.79990  .80076  .19924  .99913 

23 

38 

.80189  .80277  .19723  .99913 

22 

39 

.80388  .80476  .19524  .99912 

21 

4O 

.80585   .80674  .19326  .99911 

2O 

41 

.80782  .80872  .19128  .99910 

19 

[   42 

.80978  .81068  .18932  .99909 

18 

43 

.81173   .81264  .18736  .99909 

17 

44 

.81367  .81459  .18541  .99908 

16 

45 

.81560  .81653  .18347  .99907 

15 

46 

.81752  .81846  .18154  .99906 

14 

47 

.81944   .82038  .17962  .99905 

13 

48 

.82134  .82230  .17770  .99904 

12 

49 

.82324  .82420  .17580  .99904 

11 

50 

.82513   .82610  .17390  .99903 

1O 

51 

.82701   .82799  .17201   .99902 

9 

52 

.82888  .82987  .17013  .'99901 

8 

53 

.83075   .83175   .16825   .99900 

7 

54 

.83261   .83361   .16639  .99899 

6 

55 

.83446  .83547  .16453  .99898 

5 

56 

.83630  .83732   .16268  .99898 

4 

57 

.83813   .83916  .16084  .99897 

3 

58 

.83996  .84100  .15900  .99896 

2 

59 

.84177  .84282  .15718  .99895 

1 

52. 

.84358  .84464  .15536  .99894 

0 

/ 

SLcos  SLcot  llLtan  9Lsin 

/ 

/ 

SLsin 

SLtan 

11  Loot 

DLcos 

/ 

O 

.84  358 

.84  464 

.15  536 

.99894 

6O 

1 

.84  539 

.84646 

.15354 

.99893 

59 

2 

.84718 

.84  826 

.15  174 

.99  892 

58 

3 

.84  897 

.85006 

.14994 

.99891 

57 

4 

.85  075 

.85  185 

.14815 

.99  891 

56 

5 

.85  252 

.85  363 

.14637 

.99  890 

55 

6 

.85  429 

.85  540 

.14460 

.99889 

54 

7 

.85  605 

.85  717 

.14283 

.99888 

53 

8 

.85  780 

.85  893 

.14  107 

.99887 

52 

9 

.85  955 

.86069 

.13931 

.99886 

51 

1O 

.86  128 

.86243 

.13757 

.99885 

50 

11 

.86301 

.86417 

.13583 

.99884 

49 

12 

.86474 

.86591 

.13409 

.99883 

48 

13 

.86645 

.86  763 

.13237 

.99882 

47 

14 

.86816 

.86935 

.13065 

.99881 

46 

15 

.86987 

.87  106 

.12  894 

.99880 

45 

16 

.87156 

.87277 

.12  723 

.99  879 

44 

17 

.87325 

.87447 

.12  553 

.99  879 

43 

18 

.87494 

.87616 

.12384 

.99  878 

42 

19 

.87661 

.87  785 

.12215 

.99  877 

41 

2O 

.87  829 

.87953 

.12047 

.99876 

40 

21 

.87995 

.88  120 

.11880 

.99875 

39 

22 

.88  161 

.88287 

.11713 

.99874 

38 

23 

.88326 

.88453 

.11547 

.99  873 

37 

24 

.88490 

.88618 

.11382 

.99872 

36 

25 

.88654 

.88  783 

.11217 

.99871 

35 

26 

.88817 

.88948 

.11052 

.99  870 

34 

27 

.88980 

.89111 

.10889 

.99869 

33 

28 

.89  142 

•89274 

.10726 

.99868 

32 

29 

.89304 

.89437 

.10563 

.99867 

31 

3O 

.89464 

.89  598 

.10402 

.99866 

30 

31 

.89625 

.89  760 

.10240 

.99865 

29 

32 

.89  784 

.89920 

.10080 

.99864 

28 

33 

.89943 

.90080 

.09920 

.99863 

27 

34 

.90  102 

.90  240 

.09  760 

.99862 

26 

35 

.90260 

.90399 

.09601 

.99861 

25 

•36 

.90417 

.90557 

.09443 

.99860 

24 

37 

.90574 

.90715 

.09  285 

.99859 

23 

38 

.90  730 

.90872 

.09  128 

.99858 

22 

39 

.90885 

.91  029 

.08971 

.99857 

21 

4O 

.91  040 

.91  185 

.08815 

.99856 

2O 

41 

.91  195 

.91  340 

.08660 

.99855 

19 

42 

.91  349 

.91  495 

.08  505 

.99854 

18 

43 

.91  502 

.91  650 

.08350 

.99853 

17 

44 

.91655 

.91  803 

.08  197 

.99852 

16 

45 

.91  807 

.91957 

.08043 

.99851 

15 

46 

.91  959 

.92110 

.07  890 

.99850 

14 

47 

.92110 

.92  262 

.07  738 

.99848 

13 

48 

.92  261 

.92414 

.07  586 

.99847 

12 

49 

.92411 

.92  565 

.07  435 

.99846 

11 

5O 

.92  561 

.92716 

.07  284 

.99845 

10 

51 

.92710 

.92  866 

.07  134 

.99844 

9 

52 

.92  859 

.93  016 

.06984 

.99843 

8 

53 

.93  007 

.93  165 

.06835 

.99842 

7 

54 

.93  154 

.93313 

.06687 

.99841 

6 

55 

.93301 

.93  462 

.06538 

.99840 

5 

56 

.93  448 

.93  609 

.06391 

.99839 

4 

57 

.93  594 

.93  756 

.06244 

.99838 

3 

58 

.93  740 

.93  903 

.06097 

.99837 

2 

59 

.93  885 

.94  049 

.05951 

.99836 

1 

60 

.94030 

.94  195 

.05  805 

.99834 

O 

/ 

SLcos 

SLcot 

11  L  tan 

9Lsin 

/ 

86 


85C 


38 


/ 

SLsin  SLtan  llLcot  9Lcos 

/ 

o 

.94030  .94195   .05805   .99834 

6O 

1 

.94174  .94340  .05660  .99833 

59 

2 

.94317  .94485  .05515   .99832 

58 

3 

.94461   .94630  .05370  .99831 

57 

4 

.94603  .94773  .05227  .99830 

56 

5 

.94746  .94917  .05083   .99829 

55 

6 

.94887  .95060  .049-10  .99828 

54 

7 

.95029  .95202  .04798  .99827 

53 

8 

.95170  .95344  .04656  .99825 

52 

9 

.95310  .95486  .04514  .99824 

51 

ID 

.95450  .95627  .04373  .99823 

50 

11 

.95589  .95767  .04233  .99822 

49 

12 

.95728  .95908  .04092  .99821 

48 

13 

.95867   .96047  .03953  .99820 

47 

14 

.96005  .96187  .03813  .99819 

46 

15 

.96143  .96325   .03675   .99817 

45 

16 

.96280  .96464  .03536  .99816 

44 

17 

.96417  .96602  .03398  .99815 

43 

18 

.96553  .96739  .03261   .99814 

42 

19 

.96689  .96877  .03123  .99813 

41 

20 

.96825  .97013  .02987  .99812 

40 

21 

.96960  .97150  .02850  .99810 

39 

22 

.97095   .97285   .02715  .99809 

38 

23 

.97229  .97421   .02579  .99808 

37 

24 

.97363  .97556  .02444  .99807 

36 

25 

.97496  .97691  .02309  .99806 

35 

26 

.97629  .97825   .02175  .99804 

34 

27 

.97762  .97959  .02041  .99803 

33 

28 

.97894  .98092  .01908  .99802 

32 

29 

.98026  .98225   .01775   .99801 

31 

30 

.98157  .98358  .01642  .99800 

30 

31 

.98288  .98490  .01510  .99798 

29 

32 

.98419  .98622  .01378  .99797 

28 

33 

.98549  .98753  .01247  .99796 

27 

34 

.98679  .98884  .01116  .99795 

26 

35 

.98808  .99015  .00985  .99793 

25 

36 

.98937  .99145  .00855  .99792 

24 

37 

.99066  .99275   .00725   .99791 

23 

38 

.99194  .99405   .00595   .99790 

22 

39 

.99322  .99534  .00466  .99788 

21 

40 

.99450  .99662  .00338  .99787 

20 

41 

.99577  .99791   .00209  .99786 

19 

42 

.99704  .99919  .00081  .99785 

18 

43 

.99830  .00046  .99954  .99783 

17 

44 

.99956  .00174  .99826  .99782 

16 

45 

.00082  .00301   .99699  .99781 

15 

46 

.00207  .00427  .99573  .99780 

14 

47 

.00332  .00553   .99447  .99778 

13 

48 

.00456  .00679  .99321  .99777 

12 

49 

.00581  .00805   .99195  .99776 

11 

5O 

.00704  .00930  .99070  .99775 

1O 

51 

.00828  .01055  .98945  .99773 

9 

52 

.00951   .01179  .98821   .99772 

8 

53 

.01074  .01303   .98697  .99771 

7 

54 

.01196  .01427  .98573  .99769 

6 

55 

.01318  .01550  .98450  .99768 

5 

56 

.01440  .01673  .98327  .99767 

4 

57 

.01561   .01796  .98204  .99765 

3 

58 

.01682  .01918  .98082  .99764 

2 

59 

.01803  .02040  .97960  .99763 

1 

60 

.01923  .02162  .97838  .99761 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

/ 

9Lsin  9LtanlOLcot  9Lcos 

/ 

0 

.01923  .02162  .97838  .99761 

6O 

1 

.02043  .02283  .97717  .99760 

59 

2 

.02163   .02404  .97596  .99759 

58 

3 

.02283  .02525   .97475   .99757 

57 

4 

.02402  .02645   .97355  .99756 

56 

5 

.02520  .02766  .97234  .99755 

55 

6 

.02639  .02885  .97115  .99753 

54 

7 

.02757  .03005   .96995   .99752 

53 

8 

.02874  .03124  .96876  .99751 

52 

9 

.02992  .03242  .96758  .99749 

51 

1O 

.03109  .03361   .96639  .99748 

5O 

11 

.03226  .03479  .96521   .99747 

49 

12 

.03342  .03597  .96403  .99745 

48 

13 

.03458  .03714  .96286  .99744 

47 

14 

.03574  .03832  .96168  .99742 

46 

15 

.03690  .03948  .96052  .99741 

45 

16 

.03805   .04065   .95935   .99740 

44 

17 

.03920  .04181   .95819  .99738 

43 

18 

.04034  .04297  .95703  .99737 

42 

19 

.04149  .04413  .95587  .99736 

41 

2O 

.04262  .04528  .95472  .99734 

40 

21 

.04376  .04643  .95357  .99733 

39 

22 

.04490  .04758  .95242  .99731 

38 

23 

.04603   .04873  .95127  .99730 

37 

24 

.04715  .04987  .95013  .99728 

36 

25 

.04828  .05101   .94899  .99727 

35 

26 

.04940  .05214  .94786  .99726 

34 

27 

.05052  .05328  .94672  .99724 

33 

28 

.05164  .05441   .94559  .99723 

32 

29 

.05275  .05553  .94447  .99721 

31 

3O 

.05386  .05666  .94334  .99720 

3O 

31 

.05497  .05778  .94222  .99718 

29 

32 

.05607  .05890  .94110  .99717 

28 

33 

.05717  .06002  .93998  .99716 

27 

34 

.05827  .06113  .93887  .99714 

26 

35 

.05937  .06224  .93776  .99713 

25 

36 

.06046  .06335  .93665   .99711 

24 

37 

.06155   .06445  .93555   .99710 

23 

38 

.06264  .06556  .93444  .99708 

22 

39 

.06372  .06666  .93334  .99707 

21 

4O 

.06481  .06775  .93225  .99705 

2O 

41 

.06589  .06885   .93115  .99704 

19 

42 

.06696  .06994  .93006  .99702 

18 

43 

.06804  .07103  .92897  .99701 

17 

44 

.06911  .07211   .92789  .99699 

16 

45 

.07018  .07320  .92680  .99698 

15 

46 

.07124  .07428  .92572  .99696 

14 

47 

.07231  .07536  .92464  .99695 

13 

48 

.07337  .07643  .92357  .99693 

12 

49 

.07442  .07751  .92249  .99692 

11 

5O 

.07548  .07858  .92142  .99690 

1O 

51 

.07653  .07964  .92036  .99689 

9 

52 

.07758  .08071  .91929  .99687 

8 

53 

.07863   .08177  .91823  .99686 

7 

54 

.07968  .08283  .91717  .99684 

6 

55 

.08072  .08389  .91611  .99683 

5 

56 

.08176  .08495  .91505  .99681 

4 

57 

.08280  .08600  .91400  .99680 

3 

58 

.08383  .08705  .91295  .99678 

2 

59 

.08486  .08810  .91190  .99677 

1 

6O 

.08589  .08914  .91086  .99675 

O 

7 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

84C 


83< 


7° 


8C 


39 


/ 

9Lsin  9Ltan  10  L  cot  9Lcos 

/ 

0 

.08589  .08914  .91086  .99675 

6O 

1 

.08692  .09019  .90981  .99674 

59 

2 

.08795  .09123  .90877  .99672 

58 

3 

.08897  .09227  .90773  .99670 

57 

4 

.08999  .09330  .90670  .99669 

56 

5 

.09101  .09434  .90566  .99667 

55 

6 

.09202  .09537  .90463  .99666 

54 

7 

.09304  .09640  .90360  .99664 

53 

8 

.09405  .09742  .90258  .99663 

52 

9 

.09506  .09845  .90155  .99661 

51 

1C 

.09606  .09947  .90053  .99659 

50 

11 

.09707  .10049  .89951  .99658 

49 

12 

.09807  .10150  .89850  .99656 

48 

13 

.09907  .10252  .89748  .99655 

47 

14 

.10006  .10353  .89647  .99653 

46 

15 

.10106  .10454  .89546  .99651 

45 

16 

.10205  .10555  .89445  .99650 

44 

17 

.10304  .10656  .89344  .99648 

43 

18 

.10402  .10756  .89244  .99647 

42 

19 

.10501  .10856  .89144  .99645 

41 

2O 

.10599  .10956  .89044  .99643 

4O 

21 

.10697  .11056  .88944  .99642 

39 

22 

.10795  .11155  .88845  .99640 

38 

23 

.10893  ..11254  .88746  .99638 

37 

24 

.10990  .11353  .88647  .99637 

36 

25 

.11087  .11452  .88548  .99635 

35 

26 

-.11184  .11551  .88449  .99633 

34 

27 

.11281  .11649  .88351  .99632 

33 

28 

.11377  .11747  .88253  .99630 

32 

29 

.11474  .11845  .88155  .99629 

31 

30 

.11570  .11943  .88057  .99627 

30 

31 

.11666  .12040  .87960  .99625 

29 

32 

.11761  .12138  .87862  .99624 

28 

33 

.11857  .12235  .87765  ,.99622 

27 

34 

.11952  .12332  .87668  .99620 

26 

35 

.12047  .12428  .87572  .99618 

25 

36 

.12142  .12525  .87475  .99617 

24 

37 

.12236  .12621  .87379  .99615 

23 

38 

.12331  .12717  .87283  .99613 

22 

39 

.12425  .12813  .87187  .99612 

21 

40 

.12519  .12909  .87091  .99610 

2O 

41 

.12612  .13004  .86996  .99608 

19 

42 

.12706  .13099  .86901  .99607 

18 

43 

.12799  .13194  .86806  .99605 

17 

44 

.12892  .13289  .86711  .99603 

16 

45 

.12985  .13384  .86616  .99601 

15 

46 

.13078  .13478  .86522  .99600 

14 

47 

.13171  .13573  .86427  .99598 

13 

48 

.13263  .13667  .86333  .99596 

12 

49 

.13355  .13761  .86239  .99595 

11 

50 

.13447  .13854  .86146  .99593 

10 

51 

.13539  .13948  .86052  .99591 

9 

52 

.13630  .14041  .85959  .99589 

8 

53 

.13722  .14134  .85866  .99588 

7 

54 

.13813  .14227  .85773  .99586 

6 

55 

.13904  .14320  .85680  .99584 

5 

56 

.13994  .14412  .85588  .99582 

4 

57 

.14085  .14504  .85496  .99581 

3 

58 

.14175  .14597  .85403  .99579 

2 

59 

.14266  .14688  .85312  .99577 

1 

6O 

.14356  .14780  .85220  .99575 

0 

/ 

9  L  cos  9  L  cot  10  L  tan  9  L  sin 

/ 

/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

o 

.14356  .14780  .85220  .99575 

60 

1 

.14445  .14872  .85128  .99574 

59. 

2 

.14535  .14963  .85037  .99572 

58 

3 

.14624  .15054  .84946  .99570 

57 

4 

.14714  .15145  .84855  .99568 

56 

5 

.14803  .15236  .84764  .99566 

55 

6 

.14891  .15327  .84673  .99565 

54 

7 

.14980  .15417  .84583  .99563 

53 

8 

.15069  .15508  .84492  .99561 

52 

9 

.15157  .15598  .84402  .99559 

51 

1O 

.15245  .15688  .84312  .99557 

50 

11 

.15333  .15777  .84223  .99556 

49 

12 

.15421  .15867  .84133  .99554 

48 

13 

.15508  .15956  .84044  .99552 

47 

14 

.15596  .16046  .83954  .99550 

46 

15 

.15683  .16135  .83865  .99548 

45 

16 

.15770  .16224  .83776  .99546 

44 

17 

.15857  .16312  .83688  .99545 

43 

18 

.15944  .16401  .83599  .99543 

42 

19 

.16030  .16489  .83511  .99541 

41 

2O 

.16116  .16577  .83423  .99539 

40 

21 

.16203  .16665  .83335  .99537 

39 

22 

.16289  .16753  .83247  .99535 

38 

23 

.16374  .16841  .83159  .99533 

37 

24 

.16460  .16928  .83072  .99532 

36 

25 

.16545  .17016  .82984  .99530 

35 

26 

.16631  .17103  .82897  .99528 

34 

27 

.16716  .17190  .82810  .99526 

33 

28 

.16801  .17277  .82723  .99524 

32 

29 

.16886  .17363  .82637  .99522 

31 

3O 

.16970  .17450  .82550  .99520 

30 

31 

.17055  .17536  .82464  .99518 

29 

32 

.17139  .17622  .82378  .99517 

28 

33 

.17223  .17708  .82292  .99515 

27 

34 

.17307  .17794  .82206  .99513 

26 

35 

.17391  .17880  .82120  .99511 

25 

36 

.17474  .17965  .82035  .99509 

24 

37 

.17558  .18051  .81949  .99507 

23 

38 

.17641  .18136  .81864  .99505 

22 

39 

.17724  .18221  .81779  .99503 

21 

40 

.17807  .18306  .81694  .99501 

2O 

41 

.17800  .18391  .81609  .99499 

19 

42 

.17973  .18475  .81525  .99497 

18 

43 

.18055  .18560  .81440  .99495 

17 

44 

.28137  .18644  .81356  .99494 

16 

45 

.18220  .18728  .81272  .99492 

15 

46 

.18302  .18812  .81188  .99490 

14 

47 

.18383  .18896  .81104  .99488 

13 

48 

.18465  .18979  .81021  .99486 

12 

49 

.18547  .19063  .80937  .99484 

11 

50 

.18628  .19146  .80854  .99482 

10 

51 

.18709  .19229  .80771  .99480 

9 

52 

.18790  .19312  .80688  .99478 

8 

53 

.18871  .19395  .80605  .99476 

7 

54 

.18952  .19478  .80522  .99474 

6 

55 

.19033  .19561  .80439  .99472 

5 

56 

.19113  .19643  .80357  .99470 

4 

57 

.19193  .19725  .80275  .99468 

3 

58 

.19273  .19807  .80193  .99466 

2 

59 

.19353  .19889  .80111  .99464 

1 

60 

.19433  .19971  .80029  .99462 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

81 


40 


10 


/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

o 

.19433   .19971   .80029  .99462 

6O 

I 

.19513   .20053   .79947  .99460 

59 

2 

.19592   .20134  .79866  .99458 

58 

3 

.19672   .20216   .79784   .99456 

57 

4 

.19751   .20297  .79703   .99454 

56 

5 

.19830  .20378   .79622   .99452 

55 

6 

.19909  .20459   .79541   .99450 

54 

7 

.19988  .20540  .79460  .99448 

53 

8 

.20067   .20621    .79379  .99446 

52 

9 

.20145   .20701    .79299  .99444 

51 

1O 

.20223  .20782  .79218  .99442 

50 

11 

.20302   .20862   .79138   .99440 

49 

12 

.20380  .20942  .79058  .99438 

48 

13 

.20458  .21022  .78978  .99436 

47 

14 

.20535   .21102  .78898  .99434 

46 

15 

.20613  .21182  .78818  .99432 

45 

16 

.20691   .21  261   .78739  .99429 

44 

17 

.20768  .21341   .78659  .99427 

43 

18 

.20845   .21420  .78580  .99425 

42 

19 

.20922  .21499  .78501   .99423 

41 

20 

.20999  .21578  .78422  .99421 

4O 

21 

.21076  .21657   .78343   .99419 

39 

22 

.21153   .21736  .78264  .99417 

38 

23 

.21229  .21814  .78186  .99415 

37 

24 

.21306  .21893   .78107  .99413 

36 

25 

.21382  .21971   .78029  .99411 

35 

26 

.21458  .22049  .77951  .99409 

34 

27 

.21534  .22127  .77873  .99407 

33 

28 

.21610  .22205   .77795   .99404 

32 

29 

.21685   .22283  .77717  .99402 

31 

3O 

.21761   .22361   .77639  .99400 

3O 

31 

.21836  .22438  .77562  .99398 

29 

32 

.21912  .22516  .77484  .993.96 

28 

33 

.21987  .22593  .77407  .99394 

27 

34 

.22062  .22670  .77330  .99392 

26 

35 

.22137   .22747  .77253  .99390 

25 

36 

.22211   .22824   .77176  .99388 

24 

37 

.22286  .22901   .77099  .99385 

23 

38 

.22361   .22977  .77023  .99383 

22 

39 

.22435   .23054  .76946  .99381 

21 

40 

.22509  .23130  .76870  .99379 

20 

41 

.22583   .23206  .76794  .99377 

19 

42 

.22657   .23283  .76717  .99375 

18 

43 

.22731   .23359  .76641  .99372 

17 

44 

.22805   .23435  .76565  .99370 

16 

45 

.22878  .23510  .76490  .99368 

15 

46 

.22952  .23586  .76414  .99366 

14 

47 

.23025   .23661   .76339  .99364 

13 

48 

.23098  .23737  .76263  .99362 

12 

49 

.23171   .23812  .76188  .99359 

11 

5O 

.23244  .23887  .76113  .99357 

10 

51 

.23317  .23962  .76038  .99355 

9 

52 

.23390  .24037  -.75963  .99353 

8 

53 

.23462  .24112  .75888  .99351 

7 

54 

.23535   .24186  .75814  .99348 

6 

55 

.23607  .24261   .75739  .99346 

5 

56 

.23679  .24335   .75665   .99344 

4 

57 

.23752  .24410  .75590  .99342 

3 

58 

.23823   .24484   .75516  .99340 

2 

59 

.23895   .24558  .75442  .99337 

1 

60 

.23967  .24632  .75368  .99335 

O 

/ 

9Lcos  9Lcot  lOLtan  9Lsin 

/ 

/ 

9Lsin 

9Ltan 

1O  L  cot 

9  Lcos 

/ 

O 

.23  967 

.24  632 

.75  368 

.99335 

750 

1 

.24  039 

.24  706 

.75  294 

.99  333 

59 

2 

.24110 

.24  779 

.75  221 

.99331 

58 

3 

.24181 

.24  853 

.75  147 

.99328 

57 

4 

.24253 

.24926 

.75074 

.99  326 

56 

5 

.24324 

.25000 

.75000 

.99324 

55 

6 

.24  395 

.25  073 

.74  927 

.99322 

54 

7 

.24466 

.25  146 

.74  854 

.99319 

53 

8 

.24  536 

.25  219 

.74  781 

.99317 

52 

9 

.24607 

.25  292 

.74  708 

.99315 

51 

10 

.24677 

.25  365 

.74  635 

.99313 

5O 

11 

.24  748 

.25  437 

.74563 

.99310 

49 

12 

.24818 

.25510 

.74490 

.99308 

48 

13 

.24  888 

.25  582 

.74418 

.99306 

47 

14 

.24958 

.25  655 

.74345 

.99304 

46 

15 

.25  028 

.25  727 

.74273 

.99301 

45 

16 

.25  098 

.25  799 

.74  201 

.99299 

44 

17 

.25  168 

.25  871 

.74129 

.99  297 

43 

18 

.25  237 

.25  943 

.74057 

.99  294  i  .  42 

19 

.25307 

.26015 

.73985 

.99  292   41 

2O 

.25  376 

.26086 

.73914 

.99290  4O 

21 

.25  445 

.26158 

.73842 

.99288   39 

22 

.25514 

.26  229 

.73  771 

.99  285   38 

23 

.25  583 

.26301 

.73699 

.99  283 

37 

24 

.25  652 

.26372 

.73  628 

.99281 

36 

25 

.25  721 

.26443 

.73557 

.99278 

35 

26 

.25  790 

.26514 

.73486 

.99  276 

34 

27 

.25  858 

.26  585 

.73415 

.99274 

33 

28 

.25927 

.26655 

.73  345 

.99271 

32 

29 

.25  995 

.26  726 

.73  274 

.99269 

31 

3O 

.26063 

.26  797 

.73  203 

.99  267 

30 

31 

.26131 

.26867 

.73133 

.99  264 

29 

32 

.26  199 

.26937 

.73  063 

.99262 

28 

33 

.26267 

..27008 

.72992 

.99260 

27 

34 

.26335 

.27078 

.72  922 

.99257 

26 

35 

.26403 

.27148 

.72852 

.99255 

25 

36 

.26470 

.27218 

.72  782 

.99252 

24 

37 

.26  538 

.27  288 

.72712 

.99  250 

23 

38 

.26605 

.27357 

.72643 

.99  248 

22 

39 

.26672 

.27427 

.72573 

.99  245 

21 

40 

.26  739 

.27496 

.72  504 

.99243 

2O 

41 

.26806 

.27  566 

.72434 

.99  241 

19 

42 

.26  873 

.27635 

.72365 

.99  238 

18 

43 

.26940 

.27704 

.72  296 

.99  236 

17 

44 

.27007 

.27773 

.72227 

.99  233 

16 

45 

.27073 

.27  842 

.72155 

.99231 

15 

46 

.27140 

.27911 

.72  089 

.99  229 

14 

47 

.27  206 

.27  980 

.72020 

.99  226 

13 

48 

.27273 

.28049 

.71951 

.99  224 

12 

49 

.27339 

.28117 

.71883 

.99221 

11 

50 

.27405 

.28  186 

.71814 

.99219 

10 

51 

.27471 

.28254 

.71  746 

.99217 

9 

52 

.27537 

.28323 

.71677 

.99214 

8 

53 

.27  602 

.28391 

.71  609 

.99212 

7 

54 

.27668 

.28459 

.71  541 

.99  209 

6 

55 

.27  734 

.28  527 

.71473 

.99207 

5 

56 

.27  799 

.28  595 

.71405 

.99  204 

4 

57 

.27  864 

.28  662 

.71338 

.99  202 

3 

58 

.27930 

.28  730 

.71270 

.99  200 

2 

59 

.27995 

.28  798 

.71  202 

.99  197 

1 

60 

.28060 

.28  865 

.71  135 

.99  195 

O 

/ 

9  Lcos 

9Lcot 

lOLtan 

9Lsin 

/ 

80C 


79 


ir 


41 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

o 

.28060  .28865  .71135  .99195 

6O 

I 

.28125  .28933  .71067  .99192 

59 

2 

.28190  .29000  .71000  .99190 

58 

3 

.28254  .29067  .70933  .99187 

57 

4 

.28319  .29134  .70866  .99185 

56 

5 

.28384  .29201  .70799  .99182 

55 

6 

.28448  .29268  .70732  .99180 

54 

7 

.28512  .29335  .70665  .99177 

53 

8 

.28577  .29402  .70598  .99175 

52 

9 

.28641  .29468  .70532  .99172 

51 

1C 

.28705  .29535  .70465  .99170 

50 

11 

.28769  .29601  .70399  .99167 

49 

12 

.28833  .29668  .70332  .99165 

48 

13 

.28896  .29734  .70266  .99162 

47 

.  14 

.28960  .29800  .70200  .99160 

46 

15 

.29024  .29866  .70134  .99157 

45 

16 

.29087  .29932  .70068  .99155 

44 

17 

.29150  .29998  .70002  .99152 

43 

18 

.29214  .30064  .69936  .99150 

42 

19 

.29277  .30130  .69870  .99147 

41 

2O 

.29340  .30195  .69805  .99145 

40 

21 

.29403  .30261  .69739  .99142 

39 

22 

.29466  .30326  .69674  .99140 

38 

23 

.29529  .30391  .69609  .99137 

37 

I  24 

.29591  .30457  .69543  .99135 

36 

25 

.29654  .30522  .69478  .99132 

35 

26 

.29716  .30587  .69413  .99130 

34 

27 

.29779  .30652  .69348  .99127 

33 

28 

.29841  .30717  .69283  .99124 

32 

29 

.29903  .30782  .69218  .99122 

31 

30 

.29966  .30  846  '.69  154  .99119 

30 

31 

.30028  .30911  .69089  .99117 

29 

32 

.30090  .30975  .69025  .99114 

28 

33 

.30151  .31040  .68960  .99112 

27 

34 

.30213  .31104  .68896  .99109 

26 

35 

.30275  .31168  .68832  .99106 

25 

36 

.30336  .31233  .68767  .99104 

24 

37 

.30398  .31297  .68703  .99101 

23 

38 

.30459  .31361  .68639  .99099 

22 

39 

.30521  .31425  .68575  .99096 

21 

40 

.30582  .31489  .68511  .99093 

2O 

41 

.30643  .31552  .68448  .99091 

19 

42 

.30704  .31616  .68384  .99088 

18 

43 

.30765  .31679  .68321  .99086 

17 

44 

.30826  .31743  .68257  .99083 

16 

45 

.30887  .31806  .68194  .99080 

15 

46 

.30947  .31870  .68130  .99078 

14 

47 

.31008  .31933  .68067  .99075 

13 

48 

.31068  .31996  .68004  .99072 

12 

49 

.31129  .32059  .67941  .99070 

11 

50 

.31189  .32122  .67878  .99067 

1O 

51 

.31250  .32185  .67815  .99064 

9 

52 

.31310  .32248  .67752  .99062 

8 

53 

.31370  .32311  .67689  .99059 

7 

54 

.31430  .32373  .67627  .99056 

6 

55 

.31490  .32436  .67564  .99054 

5 

56 

.31549  .32498  .67502  .99051 

4 

57 

.31609  .32561  .67439  .99048 

3 

58 

.31669  .32623  .67377  .99046 

2 

59 

.31728  .32685  .67315  .99043 

1 

6O 

.31788  .32747  .67253  .99040 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

/ 

9Lsin 

9Ltan 

1O  L  cot 

9Lcos  -L- 

O 

.31  788 

.32  747 

.67  253 

.99040 

ou 

CO 

1 

.31847 

.32810 

.67  190 

.99038 

0  V 

2 

.31907 

.32872 

.67  128 

.99035 

58 

3 

.31  966 

.32933 

.67  067 

.99032 

57 

4 

.32025 

.32995 

.67005 

.99030 

56 

5 

.32084 

.33057 

.66943 

.99027 

55 

6 

.32  143 

.33  119 

.66881 

.99024 

54 

7 

.32  202 

.33  180 

.66  820 

.99022 

53 

CO 

8 

.32  261 

.33  242 

.66  758 

.99019 

OL 

9 

.32319 

.33303 

.66697 

.99016 

51 

1.O 

.32378 

.33  365 

.66635 

.99013 

50 

zin 

11 

.32437 

.33  426 

.66574 

.99011 

*ty 

12 

.32495 

.33487 

.66513 

.99008 

48 

13 

.32553 

.33548 

.66452 

.99005 

47 

14 

.32612 

.33609 

.66391 

.99002 

46 

15 

.32670 

.33  670 

.66330 

.99000 

45 

16 

.32  728 

.33  731 

.66  269 

.98997 

44 

17 

.32  786 

.33  792 

.66208 

.98994 

43 

18 

.32  844 

.33  853 

.66  147 

.98991 

42 

19 

.32902 

.33913 

.66087 

.98989 

41 

20 

.32960 

.33974 

.66026 

.98986 

4O 

21 

.33  018 

.34  034 

.65  966 

.98983 

39 

'22 

.33  075 

.34095 

.65  905 

.98980 

38 

1  r- 

23 

.33  133 

.34  155 

.65  845 

.98978 

o/ 

ox; 

24 

.33  190 

.34215 

.65  785 

.98975 

36 

25 

.33  248 

.34  276 

.65  724 

.98972 

35 

26 

.33  305 

.34336 

.65  664 

.98969 

34 

27 

.33  362 

.34  396 

.65  604 

.98967 

33 

28 

.33  420 

.34456 

.65  544 

.98964 

32 

29 

.33  477 

.34516 

.65  484 

.98961 

31 

30 

.33  534 

.34576 

.65  424 

.98958 

30 

31 

.33  591 

.34  635 

.65  365 

.98955 

•  29 

32 

.33647 

.34695 

.65  305 

.98953 

28 

33 

.33  704 

.34  755 

.65  245 

.98950 

27 

34 

.33  761 

.34814 

.65  186 

.98947 

26 

35 

.33  818 

.34874 

.65  126 

.98944 

25 

36 

.33  874 

.34  933 

.65  067 

.98941 

24 

37 

.33931 

.34992 

.65  008 

.98938 

23 

38 

.33  987 

.35051 

.649-19 

.98  936 

22 

39 

.34043 

.35  111 

.64  889 

.98933 

21 

40 

.34  100 

.35  170 

.64830 

.98930 

2O 

41 

.34  156 

.35  229 

,64771 

.98927 

19 

42 

.34212 

.35  288 

.64712 

.98924 

18 

43 

.34  268 

.35  347 

.64653 

.98921 

17 

44 

.34324 

.35  405 

.64595 

.98919 

16 

45 

.34  380 

.35  464 

.64  536 

.98916 

15 

46 

.34  436 

.35  523 

.64477 

.98913 

14 

47 

.34491 

.35  581 

.64419 

.98910 

13 

48 

.34  547 

.35  640 

.64360 

.9890? 

12 

49 

.34602 

.35  698 

.64302 

.98904 

11 

50 

.34658 

.35  757 

.64  243 

.98901 

1O 

51 

.34  713 

.35815 

.64  185 

.98898 

9 

52 

.34  769 

.35  873 

.64127 

.98896 

8 

53 

.34  824 

.35931 

.64069 

.98893 

7 

54 

.34  879 

.35989 

.64011 

.98890 

6 

55 

.34934 

.36047 

.63  953 

.98887 

5 

56 

.34989 

.36  105 

.63  895 

.98884 

4 

57 

.35  044 

.36  163 

.63  837 

.98  881 

3 

58 

.35  099 

.36221 

.63  779 

.98878 

2 

59 

.35  154 

.36279 

.63  721 

.98  875 

1 

60 

.35  209 

.36336 

.63  664 

.98  872 

0 

/ 

9Lccs 

9Lcot 

1O  L  tan 

9Lsin 

/ 

78C 


77C 


42 


13 


14C 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

o 

.35209  .36336  .63664  .98872 

60 

1 

.35263  .36394  .63606  .98869 

59 

2 

.35318  .36452  .63548  .98867 

58 

3 

.35373  .36509  .63491   .98864 

57 

4 

.35427  .36566  .63434  .98861 

56 

5 

.35481  .36624  .63376  .98858 

55 

6 

.35536  .36681   .63319  .98855 

54 

7 

.35590  .36738  .63262  .98852 

53 

8 

.35644  .36795   .63205  .98849 

52 

9 

.35698  .36852  .63148  .98846 

51 

1C 

.35752  .36909  .63091  .98843 

5O 

11 

.35806  .36966  .63034  .98840 

49 

12 

.35860  .37023  .62977  .98837 

48 

13 

.35914  .37080  .62920  .98834 

47 

14 

.35968  .37137  .62863  .98831 

46 

15 

.36022  .37193  .62807  .98828 

45 

16 

.36075   .37250  .62750  .98825 

44 

17 

.36129  .37306  .62694  .98822 

43 

18 

.36182  .37363   .62637  .98819 

42 

19 

.36236  .37419  .62581  .98816 

41 

20 

.36289  .37476  .62524  .98813 

40 

21 

.36342  .37532  .62468  .98810 

39 

22 

.36395  .37588  .62412  .98807 

38 

23 

.36449  .37644  .62356  .98804 

37 

24 

.36502  .37700  .62300  .98801 

36 

25 

.36555  .37756  .62244  .98798 

35 

26 

.36608  .37812  .62188  .98795 

34 

27 

.36660  .37868  .62132  .98792 

33 

28 

.36713  .37924  .62076  .98789 

32 

29 

.36766  .37980  .62020  .98786 

31 

30 

.36819  .38035  .61965   .98783 

30 

31 

.36871   .38091   .61909  .98780 

29 

32 

.36924  .38147  .61853  .98777 

28 

33 

.36976  .38202  .61798  .98774 

27 

34 

.37028  .38257  .61743  .98771 

26 

35 

.37081  .38313  .61687  .98768 

25 

36 

.37133  .38368  .61632  .98765 

24 

37 

.37185   .38423  .61577  .98762 

23 

38 

.37237  .38479  .61521.  .98759 

22 

39 

.37289  .38534  .61466  .98756 

21 

4O 

.37341   .38589  .61411  .98753 

2O 

41 

.37393  .38644  .61356  .98750 

19 

42 

.37445  .38699  .61301  .98746 

18 

43 

.37497  .38754  .61246  .98743 

17 

44 

.37549  .38808  .61192  .98740 

16 

45 

.37600  .38863  .61137  .98737 

15 

46 

.37652.  .38918  .61082  .98734 

14 

47 

.37703  .38972  .61028  .98731 

13 

48 

.37755   .39027  .60973  .98728 

12 

49 

.37806  .39082  .60918  .98725 

11 

50 

.37858  .39136  .60864  .98722 

10 

51 

.37909  .39190  .60810  .98719 

9 

52 

.37960  .39245   .60755  .98715 

8 

53 

.38011  .39299  .60701  .98712 

7 

54 

.38062  .39353  .60647  .98709 

6 

55 

.38113  .39407  .60593  .98706 

5 

56 

.38164  .39461   .60539  '.98703 

4 

57 

.38215  .39515   .60485  .98700 

3 

58 

.38266  .39569  .60431   .98697 

2 

59 

.38317  .39623  .60377  .98694 

1 

60 

.38368  .39677  .60323  .98690 

O 

/ 

9Lcos  9LcotlOLtan9Lsin 

/ 

/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

~o 

.38368  .39677  .60323  .98690 

6O 

1 

.38418  .39731   .60269  .98687 

59 

2 

.38469  .39785   .60215  .98684 

58 

3 

.38519  .39838  .60162  .98681 

57 

4 

.38570  .39892  .60108  .98678 

56 

5 

.38620  .39945   .60055   .98675 

55 

6 

.38670  .39999  .60001  .98671 

54 

7 

.38721  .40052  .59948  .98668 

53 

8 

.38771  .40106  .59894  .98665 

52 

9 

.38821  .40159  .59841   .98662 

51 

10 

.38871   .40212  .59788  .98659 

5O 

11 

.38921  .40266  .59734  .98656 

49 

12 

.38971   .40319  .59681   .98652 

48 

13 

.39021   .40372  .59628  .98649 

47 

14 

.39071   .40425   .59575   .98646 

46 

15 

.39121  .40478  .59522  .98643 

45 

16 

.39170  .40531  .59469   .98640 

44 

17 

.39220  .40584  .59416  .98636 

43 

18 

.39270  .40636  .59364  .98633 

42 

19 

.39319  .40689  .59311   .98630 

41 

20 

.39369  .40742  .59258  .98627 

40 

21 

.39418  .40795   .59205   .98623 

39 

22 

.39467  .40847  .59153  .98620 

38 

23 

.39517  .40900  .59100  .98617 

37 

24 

.39566  .40952  .59048  .98614 

36 

25 

.39615  .41005  .58995  .98610 

35 

26 

.39.664  .41057  .58943  .98607 

34 

27 

.39713  .41109  .58891   .98604 

33 

28 

.39762  .41161   .58839  .98601 

32 

29 

.39811  .41214  .58786  .98597 

31 

3O 

.39860  .41266  .58734  .98594 

30 

31 

.39909  .41318  .58682  .98591 

29 

32 

.39958  .41370  .58630  .98588 

28 

33 

.40006  .41422  .58578  .98584 

27 

34 

.40055   .41474  .58526  .98581 

26 

35 

.40103  .41526  .58474  .98578 

25 

36 

.40152  .41578  .58422  .98574 

24 

37 

.40200  .41629  .58371   .98571 

23 

38 

.40249  .41681   .58319  .98568 

22 

39 

.40297  .41733  .58267  .98565 

21 

4O 

.40346  .41784  .58216  .98561 

20 

41 

.40394  .41836  .58164  .98558 

19 

42 

.40442  .41887  .58113  .98555 

18 

43 

.40490  .41939  .58061  .98551 

17 

44 

.40538  .41990  .58010  .98548 

16 

45 

.40586  .42041  .57959  .98545 

15 

46 

.40634  .42093  .57907  .98541 

14 

47 

.40682  .42144  .57856  .98538 

13 

48 

.40730  .42195  .57805   .98535 

12 

49 

.40778  .42246  .57754  .98531 

11 

50 

.40825   .42297  .57703  .98528 

1O 

51 

.40873  .42348  .57652  .98525 

9 

52 

.40921  .42399  .57601  .98521 

8 

53 

.40968  .42450  .57550  .98518 

7 

54 

.41016  .42501  .57499  .98515 

6 

55 

.41063  .42552  .57448  .98511 

5 

56 

.41111  .42603  .57397  .98508 

4 

57 

.41158  .42653  .57347  .98505 

3 

58 

.41205  .42704  .57296  .98501 

2 

59 

.41252  .42755  .57245  .98498 

1 

60 

.41300  .42805  .57195   .98494 

O 

/ 

9Lcos  9LcotlOLtan  9Lsin 

/ 

76' 


75C 


15' 


16 


43 


/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

o 

.41300  .42805  .57195   .98494 

60 

I 

.41347  .42856  .57144  .98491 

59 

2 

.41394  .42906  .57094  .98488 

58 

3 

.41441   .42957  .57043   .98484 

57 

4 

.41488  .43007  .56993  .98481 

56 

5 

.41535   .43057  .56943   .98477 

55 

6 

.41582  .43108  .56892  .98474 

54 

7 

.41628  .43158  .56842  .98471 

53 

8 

.41675   .43208  .56792  .98467 

52 

9 

.41722  .43258  .56742  .98464 

51 

10 

.41768  .43308  .56692  .98460 

50 

11 

ATSIS   .43358  .56642   .98457 

49 

12 

.41861    .43408  .56592  .98453 

48 

13 

.41908  .43458  .56542  .98450 

47 

14 

.41954  .43508  .56492  .98447 

46 

15 

.42001   .43558  .56442  .98443 

45 

16 

.42047   .43607  .56393  .98440 

44 

17 

.42093  .43657  .56343   .98436 

43 

18 

.42140  .43707  .56293   .98433 

42 

19 

.42186  .43756  .56244  .98429 

41 

2O 

.42232  .43806  .56194   .98426 

4O 

21 

.42278  .43855   .56145   .98422 

39 

22 

.42324  .43905   .56095  .98419 

38 

23 

.42370  .43954  .56046  .98415 

37 

24 

.42416  .44004  .55996  .98412 

36 

25 

.42461  .44053  .55947  .98409 

35 

26 

.42507  .44102  .55898  .98405 

34 

27 

.42553  .44151   .55849  .98402 

33 

28 

.42599  .44201  .55799  .98398 

32 

29 

.42644  .44250  .55750  .98395 

31 

3O 

.42690  .44299  .55701   .98391 

30 

31 

.42735  .44348  .55652   .98388 

29 

32 

.42781  .44397  .55603   .98384 

28 

33 

.42-826  .44446  .55554  .98381 

27 

34 

.42872  .44495   .55505   .98377 

26 

35 

.42917  .44544  .55456  .98373 

25 

36 

.42962  .44592  .55408  .98370 

24 

37 

.43008  .44641    .55359  .98366 

23 

38 

.43053  .44690  .55310  .98363 

22 

39 

.43098  .44738  .55262  .98359 

21 

4O 

.43143   .44787  .55213   .98356 

2O 

41 

.43188  .44836  .55164  .98352 

19 

42 

.43233  .44884  .55116  .98349 

18 

43 

.43278  .44933  .55067  .98345 

17 

44 

.43323  .44981   .55019  .98342 

16 

45 

.43367  .45029  .54971   .98338 

15 

46 

.43412  .45078  .54922   .98334 

14 

47 

.43457  .45126  .54874  .98331 

13 

48 

.43502  .45174  .54826  .98327 

12 

49 

.43546  .45222  .54778  .98324 

11 

50 

.43591  .45271  .54729  .98320 

10 

51 

.43635  .45319  .54681   .98317 

9 

52 

.43680  .45367  .54633   .98313 

8 

53 

.43724  .45415   .54585   .98309 

7 

54 

.43769  .45463  .54537  .98306 

6 

55 

.43813  .45511   .54489  .98302 

5 

56 

.43857  .45559  .54441  .98299 

4 

57 

.43901   .45606  .54394  .98295 

3 

58 

.43946  .45654  .54346  .98291 

2 

59 

.43990  .45702  .54298  .98288 

1 

6O 

.44034  .45750  .54250  .98284 

O 

/ 

9  L  cos  9  L  cot  10  L  tan  9  L  sin 

/ 

/ 

9  Lain 

9Ltan 

10  L  cot 

9Lcos 

/ 

0 

.44034 

.45  750 

.54250 

.98284 

60 

1 

.44  078 

.45  797 

.54203 

.98  281 

59 

2 

.44  122 

.45  845 

.54  155 

.98277 

58 

3 

.44  166 

.45  892 

.54  108 

.98  273 

57 

4 

.44  210 

.45  940 

.54060 

.98  270 

56 

5 

.44  253 

.45  987 

.54013 

.98266 

55 

6 

.44  297 

.46035 

.53  965 

.98  262 

54 

7 

.44341 

.46082 

.53918 

.98  259 

53 

8 

.44385 

.46  130 

.53870 

.98255 

52 

9 

.44428 

.46  177 

.53823 

.98251 

51 

1O 

.44472 

.46224 

.53  776 

.98248 

50 

11 

.44516 

.46271 

.53  729 

.98244 

49 

12 

.44559 

.46319 

.53681 

.98240 

48 

13 

.44602 

.46366 

.53  634 

.98237 

47 

14 

.44646 

.46413 

.53587 

.98233 

46 

15 

.44  689 

.46  460 

.53  540 

.98  229 

45 

16 

.44  733 

.46  507 

.53  493 

.98  226 

44 

17 

.44  776 

.46554 

.53  446 

.98  222 

43 

18 

.44819 

.46601 

.53  399 

.98  218 

42 

19 

.44862 

.46648 

.53  352 

.98215 

41 

2O 

.44905 

.46694 

.53  306 

.98211 

40 

21 

.44  948 

.46741 

.53259 

.98  207 

39 

22 

.44992 

.46  788 

.53212 

.98  204 

38 

23 

.45  035 

.46  835 

.53165 

.98  200 

37 

24 

.45  077 

.46881 

.53119 

.98  196 

36 

25 

.45  120 

.46928 

.53072 

.98  192 

35 

26 

.45  163 

.46975 

.53  025 

.98  189 

34 

27 

.45  206 

.47021 

.52979 

.98  185 

33 

28 

.45  249 

.47068 

.52932 

.98181 

32 

29 

.45  292 

.47  114 

.52886 

.98177 

31 

30 

.45  334 

.47  160 

.52840 

.98174 

30 

31 

.45377 

.47  207 

.52  793 

.98170 

29 

32 

.45  419 

.47  253 

.52  747 

.98  166 

28 

33 

.45  462 

.47  299 

.52701 

.98  162 

27 

34 

.45  504 

.47346 

.52654 

.98159 

26 

35 

.45  547 

.47  392 

.52608 

.98  155 

25 

36 

.45  589 

.47438 

.52562 

.98151 

24 

37 

.45  632 

.47  484 

.52516 

.98  147 

23 

38 

.45  674 

.47530 

.52470 

.98  144 

22 

39 

.45  716 

.47576 

.52424 

.98  140 

21 

40 

.45  758 

.47  622 

.52378 

.98  136 

2O 

41 

.45  801 

.47  668 

.52332 

.98  132 

19 

42 

.45  843 

.47714 

.52  286 

.98  129 

18 

43 

.45  885 

.47  760 

.52  240 

.98  125 

17 

44 

.45  927 

.47806 

.52  194 

.98  121 

16 

45 

.45969 

.47  852 

.52  148 

.98117 

15 

46 

.46011 

.47  897 

.52  103 

.98113 

14 

47 

.46053 

.47  943 

.52057 

.98110 

13 

48 

.46095 

.47  989 

.52011 

.98  106 

12 

49 

.46  136 

.48035 

.51965 

.98  102 

11 

50 

.46178 

.48080 

.51920 

.98098 

1O 

51 

.46  220 

.48  126 

.51874 

.98094 

9 

52 

.46262 

.48  171 

.51829 

.98090 

8 

53 

.46303 

.48217 

.51783 

.98087 

7 

54 

.46345 

.48  262 

.51  738 

.98083 

6 

55 

.46386 

.48307 

.51693 

.98079 

5 

56 

.46428 

.48353 

.51647 

.98075 

4 

57 

.46469 

.48398 

.51  602 

.98071 

3 

58 

.46511 

.48443 

.51557 

.98067 

2 

59 

.46552 

.48489 

.51511 

.98063 

1 

60 

.46594 

.48  534 

.51466 

.98060 

O 

/ 

9Lcos 

9Lcot 

1O  L  tan 

91  sin 

/ 

73° 


44 


17 


18C 


/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

o 

.46594  .48534  .51466  .98060 

6O 

1 

.46635  .48579  .51421  .98056 

59 

2 

.46676  .48624  .51376  .98052 

58 

3 

.46717  .48669  .51331   .98048 

57 

4 

.46758  .48714  .51286  .98044 

56 

5 

.46800  .48759  .51241   .98040 

55 

6 

.46841   .48804  .51196  .98036 

54 

7 

.46882  .48849  .51151  .98032 

53 

•8 

.46923  .48894  .51106  .98029 

52 

9 

.46964  .48939  .51061  .98025 

51 

10 

.47005  .48984  .51016  .98021 

5O 

11 

.47045   .49029  .50971   .98017 

49 

12 

.47086  .49073  .50927  .98013 

48 

13 

.47127  .49118  .50882  .98009 

47 

14 

.47168  .49163  .50837  .98005 

46 

15 

.47209  .49207  .50793  .98001 

45 

16 

.47249  .49252  .50748  .97997 

44 

17 

.47290  .49296  .50704  .97993 

43 

18 

.47330  .49341  .50659  .97989 

42 

19 

.47371  .49385  .50615  .97986 

41 

20 

.47411  .49430  .50570  .97982 

40 

21 

.47452  .49474  .50526  .97978 

39 

22 

.47492  .49519  .50481   .97974 

38 

23 

.47533  .49563  .50437  .97970 

37 

24 

.47573  .49607  .50393  .97966 

36 

25 

.47613  .49652  .50348  .97962 

35 

26 

.47654  .49696  .50304  .97958 

34 

27 

.47694  .49740  .50260  .97954 

33 

28 

.47734  .49784   .50216  .97950 

32 

29 

.47774  .49828  .50172  .97946 

31 

30 

.47814  .49872  .50128  .97942 

30 

31 

.47854  .49916  .50084  .97938 

29 

32 

.47894  .49960  .50040  .97934 

28 

33 

.47934  .50004  .49996  .97930 

27 

34 

.47974  .50048  .49952  .97926 

26 

35 

.48014  .50092  .49908  .97922 

25 

36 

.48054  .50136  .49864  .97918 

24 

37 

.48094  .50180  .49820  .97914 

23 

38 

.48133  .50223  .49777  .97910 

22 

39 

.48173  .50267  .49733   .97906 

21 

4O 

.48213  .50311  .49689  .97902 

2O 

41 

.48252  .50355  .49645   .97898 

19 

42 

.48292  .50398  .49602  .97894 

18 

43 

.48332  .50442  .49558  .97890 

17 

44 

.48371  .50485  .49515  .97886 

16 

45 

.48411  .50529  .49471   .97882 

15 

46 

.48450  .50572  .49428  .97878 

14 

47 

.48490  .50616  .49384  .97874 

13 

48 

.48529  .50659  .49341  .97870 

12 

49 

.48568  .50703  .49297  .97866 

11 

50 

.48607  .50746  .49254  .97861 

1O 

51 

.48647  .50789  .49211  .97857 

9 

52 

.48686  .50833  .49167  .97853 

8 

53 

.48725  .50876  .49124  .97849 

7 

54 

.48764  .50919  .49081  .97845 

6 

55 

.48803  .50962  .49038  .97841 

5 

56 

.4S8H2  .51005  .48995   .97837 

4 

57 

.48881  .51048  .48952  .97833 

3 

58 

.48920  .51092  .48908  .97829 

2 

59 

.48959  .51135   .48865   .97825 

1 

60 

.48998  .51  178  .48822  .97821 

0 

/ 

9Lcos  9Lcot  lOLtan  9Lsin 

/ 

/ 

9Lsin  9Ltan  10  L  cot  9Lcos 

/ 

O 

.48998  .51178  .48822  .97821 

6O 

1 

.49037  .51221  .48779  .97817 

59 

2 

.49076  .51264  .48736  .97812 

58 

3 

.49115   .51306  .48694  .97808 

57 

4 

.49153  .51349  .48651   .97804 

56 

5 

.49192   .51392  .48608  .97800 

55 

6 

.49231   .51435   .48565  .97796 

54 

7 

.49269   .51478  .48522  .97792 

53 

8 

.49308  .51520  .48480  .97788 

52 

9 

.49347   .51563  .48437  .97784 

51 

1O 

.49385   .51606  .48394  .97779    5O 

11 

.49424   .51648  .48352  .97775 

49 

12 

.49462   .51691  .48309  .97771 

48 

13 

.49500  .51734  .48266  .97767 

47 

14 

.49539  .51776  .48224  .97763 

46 

15 

.49577  .51819  .48181  .97759 

45 

16 

.49615   .51861   .48139  .97754 

44 

17 

.49654  .51903   .48097  .97750 

43 

18 

.49692  .51946  .48054  .97746 

42 

19 

.49730  .51988  .48012  .97742 

41 

2O 

.49768  .52031   .47969  .97738 

40 

21 

.49806  .52073   .47927  .97734 

39 

22 

.49844  .52115   .47885  .97729 

38 

-    23 

.49882  .52157   .47843  .97725 

37 

24 

.49920  .52200  .47800  .97721 

36 

25 

.49958  .52242  .47758  .97717 

35 

26 

.49996  .52284  .47716  .97713 

34 

27 

.50034   .52326  .47674  .97708 

33 

28 

.50072   .52368  .47632  .97704 

32 

29 

.50110  .52410  .47590  .97700 

31 

30 

.50148  .52452  .47548  .97696 

3O 

31 

.50185,  .52494  .47506  .97691 

29 

32 

.50223   .52536  .47464  .97687 

28 

33 

.50261   .52578  .47422  .97683 

27 

34 

.50298  .52620  .47380  .97679 

26 

35 

.50336  .52661   .47339  .97674 

25 

36 

.50374  .52703  .47297  .97670 

24 

37 

.50411  .52745  .47255   .97666 

23 

38 

.50449  .52787  .47213  .97662 

22 

39 

.50486  .52829  .47171  .97657 

21 

4O 

.50523  .52870  .47130  .97653 

20 

41 

.50561   .52912  .47088  .97649 

19 

42 

.50598  .52953  .47047  .97645 

18 

43 

.50635  .52995   .47005   .97540 

17 

44 

.50673  .53037  .46963  .97636 

16 

45 

.50710  .53078  .46922   .97632 

15 

46 

.50747  .53120  .46880  .97628 

14 

47 

.50784  .53161   .46839  .97623 

13 

48 

.50821   .53202  .46798  .97619 

12 

49 

.50858  .53244    .46756  .97615 

11 

50 

.50896  .53285  .46715   .97610 

10 

51 

.50933  .53327  .46673  .97606 

9 

52 

.50970  .53368  .46632  .97602 

8 

53 

.51007  .53409  .46591  .97597 

7 

54 

.51043   .53450  .46550  .97593 

6 

55 

.51080  .53492  .46508  .97589 

5 

56 

.51117  .53533  .46467  .97584 

4 

57 

.51154  .53574  .46426  .97580 

3 

58 

.51191   .53615  .46385   .97576 

2 

59 

.51227  .53656  .46344  .97571 

1 

60 

.51264  .53697  .46303  .97567 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

72C 


71 


19 


20C 


45 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

o 

.51264  .53697^.46303  .97567 

6O 

1 

.51301   .53738  .46262  .97563 

59 

2 

.51338  .53779  .46221  .97558 

58 

3 

.51374  .53820  .46180  .97554 

57 

4 

.51411  .53861  .46139  .97550 

56 

5 

.51447  .53902  .46098  .97545 

55 

6 

.51484  .53943  .46057  .97541 

54 

7 

.51520  .53984  .46016  .97536 

53 

8 

.51557  .54025  .45975  .97532 

52 

9 

.51593  .54065  .45935  .97528 

51 

1O 

.51629  .54106  .45894  .97523 

5O 

11 

.51666  .54147  .45853  .97519 

49 

12 

.51702  .54187  .45813  .97515 

48 

13 

.51738  .54228  .45772  .97510 

47 

14 

.51774  .54269  .45731   .97506 

46 

15 

.51811  .54309  .45691    97501 

45 

16 

.51847  .54350  .45650  .97497 

44 

17 

.51883  .54390  .45610  .97492 

43 

18 

.51919  .54431  .45569  .97488 

42 

19 

.51955  .54471  .45529  .97484 

41 

2O 

.51991  .54512  .45488  .97479 

4O 

21 

.52027  .54552  .45448  .97475, 

39 

22 

.52063  .54593  .45407  .97470 

38 

23 

.52099  .54633  .45367  .97466 

37 

24 

.52135   .54673  .45327  .97461 

36 

25 

.52171  .54714  .45286  .97457 

35 

26 

.52207  .54754  .45246  .97453 

34 

27 

.52242  .54794  .45206  .97448 

33 

28 

.52  278  .54  835   .45  165  .97  444 

32 

29 

.52314  .54875  .45125   .97439 

31 

3O 

.52350  .54915  .45085   .97435 

3Q 

31 

.52385  .54955  .45045  .97430 

29 

32 

.52421  .54995   .45005   .97426 

28 

33 

.52456  .55035   .44965   .97421 

27 

34 

.52492  .55075   .44925  .97417 

26 

35 

.52527  .55115   .44885  .97412 

25 

36    .52563  .55155   .44845   .97408 

24 

37    .52598  .55195   .44805   .97403 

23 

*3S 

.52634  .55235  .44765   .97399 

22 

39 

.52669  .55275   .44725  .97394 

21 

4O    .52705   .55315   .44685   .97390 

2O 

41    .52740  .55355   .44645  .97385 

19 

42 

.52775   .55395   .44605   .97381 

18 

43 

.52811  .55434  .44566  .97376 

17 

44 

.52846  .55474  .44526  .97372 

16 

45 

.52881   .55514  .44486  .97367 

15 

46 

.52916  .55554  .44446  .97363 

14 

47 

.52951  .55593  .44407  .97358 

13 

48 

.52986  .55633  .44367  .97353 

12 

49 

.53021  .55673  .44327  .97349 

11 

5O 

.53056  .55712  .44288  .97344 

1O 

51 

.53092   .55752  .44248  .97340 

9 

52 

.53126  .55791   .44209  .97335 

8 

53 

.53161  .55831   .44169  .97331 

7 

54 

.53196  .55870  .44130  .97326 

6 

55 

.53231   .55910  .44090  .97322 

5 

56 

.53266  .55949  .44051  .97317 

4 

57 

.53301   .55989  .44011  .97312 

3 

58 

.53336  .56028  .43972  .97308 

2 

59 

.53370  .56067  .43933  .97303 

1 

60 

.53405  .56107  .43893  .97299 

O 

/ 

9Lcos  9Lcot  lOLtan  9Lsin 

/ 

/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

O 

.53405  .56107  .43893  .97299 

6O 

1 

.53440  .56146  .43854  .97294 

59 

2 

.53475  .56185  .43815  .97289 

58 

3 

.53509  .56224  .43776  .97285 

57 

4 

.53544  .56264  .43736  .97280 

56 

5 

.53578  .56303  .43697  .97276 

55 

6 

.53613  .56342  .43658  .97271 

54 

7 

.53647  .56381  .43619  .97266 

53 

8 

.53682  .56420  .43580  .97262 

52 

9 

.53716  .56459  .43541  '97257 

51 

1O 

.53751  .56498  .43502  .97252 

5O 

11 

.53785  .56537  .43463  .97248 

49 

12 

.53819  .56576  .43424  .97243 

48 

13 

.53854  .56615  .43385  .97238 

47 

14 

.53888  .56654  .43346  .97234 

46 

15 

.53922  .56693  .43307  .97.229 

45 

16 

.53957  .56732  .43268  .97224 

44 

17 

.53991  .56771  .43229  .97220 

43 

18 

.54025  .56810  .43190  .97215 

42 

19 

.54059  .56849  .43151  .97210 

41 

20 

.54093  .56887  .43113  .97206 

4O 

21 

.54127  .56926  .43074  .97201 

39 

22 

.54161  .56965  .43035  .97196 

38 

23 

.54195  .57004  .42996  .97192 

37 

24 

.54229  .57042  .42958  .97187 

36 

25 

.54263  .57081  .42919  .97182 

35 

26 

.54297  .57120  .42880  .97178 

34 

27 

.54331  .57158  .42842  .97173 

33 

28 

.54365  .57197  .42803  .97168 

32  1 

29 

.54399  .$7235  .42765  .97163 

31 

30 

.54433  .57274  .42726  .97159 

30 

31 

.54466  .57312  .42688  .97154 

29 

32 

.54500  .57351  .42649  .97149 

28 

33 

.54534  .57389  .42611  .97145 

27 

34 

.54567  .57428  .42572  .97140 

26 

35 

.54601  .57466  .42534  .97135 

25 

36 

.54635  .57504  .42496  .97130 

24 

37 

.54668  .57543  .42457  .97126 

23 

38 

.54702  .57581  .42419  .97121 

22 

39 

.54735  .57619  .42381  .97116 

21 

4O 

.54769  .57658  .42342  .97111 

2O 

41 

.54802  .57696  .42304  .97107 

19 

42 

.54836  .57734  .42266  .97102 

18 

43 

.54869  .57772  .42228  .97097 

17 

44 

.54903  .57810  .42190  .97092 

16 

45 

.54936  .57849  .42151  .97087 

15 

46 

.54969  .57887  .42113  .97083 

14 

47 

.55003  .57925  .42075  .97078 

13 

48 

.55036  .57963  .42037  .97073 

12 

49 

.55069  .58001  .41999  .97068 

11 

50 

.55102  .58039  .41961  .97063 

10 

51 

.55136  .58077  .41923  .97059 

9 

52 

.55169  .58115  .41885  .97054 

8 

53 

.55202  .58153  .41847  .97049 

7 

54 

.55235  .58191  .41809  .97044 

6 

55 

.55268  .58229  .41771  .97039 

5 

56 

.55301  .58267  .41733  .97035 

4 

57 

.55334  .58304  .41696  .97030 

3 

58 

.55367  .58342  .41658  .97025 

2 

59 

.55400  .58380  .41620  .97020 

1 

60 

.55433  .58418  .41582  .97015 

O 

/ 

9Lcos  9  Loot  lOLtan  9Lsin 

/ 

70C 


69 


22' 


/ 

9Lsin 

9Ltan 

1O  L  cot 

9  Lcos 

/ 

o 

.55433 

.58418 

.41  582 

.97015 

6O 

1 

.55  466 

.58455 

.41545 

.97010 

59 

2 

.55  499 

.58493 

.41507 

.97  005 

58 

3 

.55  532 

.58531 

.41  469 

.97  001 

57 

4 

.55  564 

.58  569 

.41431 

.96996 

56 

5 

.55  597 

.58606 

.41  394 

.96991 

55 

6 

.55  630 

.58644 

.41  356 

.96986 

54 

7 

.55  663 

.58681 

.41319 

.96981 

53 

8 

.55  695 

.58719 

.41  281 

.96976 

52 

9 

.55  728 

.58757 

.41  243 

.96971 

51 

10 

.55  761 

.58  794 

.41  206 

.96966 

5O 

11 

.55  793 

.58832 

.41  168 

.96962 

49 

12 

.55  826 

.58869 

.41  131 

.96957 

48 

13 

.55  858 

.58907 

.41  093 

.96952 

47 

14 

.55  891 

,58944 

.41  056 

.96947 

46 

15 

.55.923 

.58981 

.41  019 

.96942 

45 

16 

.55  956 

.59019 

.40981 

.96937 

44 

17 

.55  988 

.59056 

.40944 

.96932 

43 

18 

.56021 

.59094 

.40906 

.96927 

42 

19 

.56053 

.59131 

.40869 

.96922 

41 

20 

.56085 

.59168 

.40832 

.96917 

40 

21 

.56118 

.59205 

.40  795 

.96912 

39 

22 

.56150 

.59243 

.40757 

.96907 

38 

23 

.56182 

.59280 

.40  720 

.96903 

37 

24 

.56215 

.59317 

.40683 

.96898 

36 

25 

.56247 

.59354 

.40646 

.96893 

35 

26 

.56279 

.59391 

.40  609 

.96888 

34 

27 

.56311 

.59429 

.40571 

.96883 

33 

28 

.56343 

.59466 

.40534 

.96878 

32 

29 

.56375 

.59503 

.40497 

.96873 

31 

30 

.56408 

.59540 

.40460 

.96868 

30 

31 

.56440 

.59577 

.40423 

.96  863 

29 

32 

.56472 

.59614 

.40386 

.96  858 

28 

33 

.56504 

.59651 

.40349 

.96853 

27 

34 

.56536 

.59688 

.40312 

.96848 

26 

35 

.56568 

.59725 

.40275 

.96843 

25 

36 

.56  599 

.59  762 

.40  238 

.96838 

24 

37 

.56631 

.59  799 

.40  201 

.96833 

23 

38 

.56663 

.59835 

.40  165 

.96828 

22 

39 

.56695 

.59872 

.40  128 

.96823 

21 

40 

.56727 

.59909 

.40091 

.96818 

20 

41 

.56759 

.59946 

.40054 

.96813 

19 

42 

.56  790 

.59983 

.40017 

.96808 

18 

43 

.56822 

.60019 

.39981 

.96803 

17 

44 

.56854 

.60056 

.39944 

.96  798 

16 

45 

.56886 

.60093 

.39907 

.96793 

15 

46 

.56917 

.60  130 

.39870 

.96  788 

14 

47 

.56949 

.60  166 

.39834 

.96  783 

13 

48 

.56980 

.60  203 

.39  797 

.96778 

12 

49 

.57012 

.60240 

.39  760 

.96772 

11 

50 

.57044 

.60276 

.39  724 

.96  767 

10 

51 

.57075 

.60313 

.39687 

.96  762 

9 

52 

.57107 

.60349 

.39651 

.96757 

8 

53 

.57138 

.60386 

.39614 

.96752 

7 

54 

.57169 

.60422 

.39578 

.96  747 

.  6 

55 

.57201 

.60459 

.39  541 

.96  742 

5 

56 

.57232 

.60495 

.39505 

.96737 

4 

"57 

.57264 

.60532 

.39468 

.96732 

3 

58 

.57295 

.60568 

.39432 

.96  727 

2 

59 

.57326 

.60605 

.39395 

.96  722 

1 

60 

.57358 

.60641 

.39359 

.96  717 

O 

/ 

9Lcos 

9LcotlOLtan9Lsin 

/ 

/ 

9Lsin 

9Ltan 

1O  L  cot 

9  Lcos 

/ 

O 

.57358 

.60641 

.39359 

.96717 

6O 

1 

.57389 

.60677 

.39323 

.96711 

59 

2 

.57420 

.60  714 

.39286 

.96  706 

58 

3 

.57451 

.60  750 

.39250 

.96  701 

57 

4 

.57482 

.60786 

.39  214 

.96696 

56 

5 

.57514 

.60823 

.39177 

.96691 

55 

6 

.57545 

.60859 

.39  141 

.96686 

54 

7 

.57576 

.60895 

.39  105 

.96681 

53 

8 

.57607 

.60931 

.39069 

.96676 

52 

9 

.57638 

.60967 

.39033 

.96670 

51 

1O 

.57669 

.61  004 

.38996 

.96665 

50 

11 

.57700 

.61  040 

.38960 

.96660 

49 

12 

.57731 

.61  076 

.38924 

.96655 

48 

13 

.57762 

.61]  12 

.38  888 

.96650 

47 

14 

.57793 

.61  148 

.38852 

.96645 

46 

15 

.57824 

.61  184 

.38816 

.96640 

45 

16 

.57855 

.61  220 

.38  780 

.96634 

44 

17 

.57885 

.61  256 

.38  744 

.96629 

43 

18 

.57916 

.61  292 

.38  708 

.96624 

42 

19 

.57947 

.61  328 

.38672 

.96619 

41 

20 

.57978 

.61  364 

.38636 

.96614 

40 

21 

.58008 

.61  400 

.38  600 

.96608 

39 

22 

.58039 

.61436 

.38  564 

.96603 

38 

23 

.58070 

.61  472 

.38528 

.96  598 

37 

24 

.58101 

.61  508 

.38492 

.96593 

36 

25 

.58131 

.61  544 

.38456 

.96  588 

35 

26 

.58  162 

.61  579 

.38421 

.96  582 

34 

27 

.58  192 

.61  615 

.38385 

.96577 

33 

28 

.58223 

.61651 

.38349 

.96572 

32 

29 

.58253 

.61  687 

.38313 

.96567 

31 

.30 

.58284 

.61  722 

.38278 

.96562 

30 

31 

.58314 

.61  758 

.38  242 

.96556 

29 

32 

,58345 

.61  794 

.38  206 

.96551 

28 

33 

.58375 

.61  830 

.38  170 

.96546 

27 

34 

.58406 

.61  865 

.38  135 

.96541 

26 

35 

.58436 

.61901 

.38099 

.96535 

25 

36 

.58467 

.61936 

.38064 

.96530 

24 

37 

.58497 

.61  972 

.38028 

.96  525 

23 

38 

.58527 

.62008 

.37992 

.96  520 

22 

39 

.58557 

.62043 

.37957 

.96514 

21 

4O 

.58588 

.62079 

.37921 

.96509 

2O 

41 

.58618 

.62  114 

.37886 

.96504 

19 

42 

.58648 

.62  150 

.37850 

.96498 

18 

43 

.58678 

.62185 

.37815 

.96493 

17 

44 

.58  709 

.62221 

.37779 

.96488 

16 

45 

.58739 

.62256 

.37  744 

.96483 

15 

46 

.58769 

.62  292 

.37  708 

.96477 

14 

47 

.58  799 

.62327 

.37673 

.96472 

13 

48 

.58829 

.62  362 

.37638 

.96467 

12 

49 

.58859 

.62398 

.37602 

.96461 

11 

5O 

.58889 

.62433 

.37567 

.96456 

10 

51 

.58919 

.62  468 

.37532 

.96451 

9 

52 

.58949 

.62  504 

.37496 

.96445 

8 

53 

.58979 

.62  539 

.37461 

.96440 

7 

54 

.59009 

.62  574 

.37426 

.96435 

6 

55 

.59039 

.62  609 

.37391 

.96429 

5 

56 

.59069 

.62645 

.37355 

.96424 

4 

57 

.59098 

.62680 

.37320 

.96419 

3 

58 

.59128 

.62715 

.37  285 

.96413 

2 

59 

.59158 

.62  750 

.37  250 

.96408 

1 

60 

.59188 

.62  785 

.37215 

.96403 

0 

/ 

9  Lcos 

9  L  cot  1O  L  tan 

9  L  sin   / 

68' 


67° 


23' 


47 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

0 

.59188  .62785  .37215  .96403 

6O 

1 

.59218  .62820  .37180  .96397 

59 

2 

.59247  .62855   .37145   .96392 

58 

3 

.59277  .62890  .37110  .96387 

57 

4 

.59307  .62926  .37074  .96381 

56 

5 

.59336  .62961  .37039  .96376 

55 

6 

.59366  .62996  .37004  .96370 

54 

7 

.59396  .63031   .36969  .96365 

53 

8 

.59425   .63066  .36934  .96360 

52 

9 

.59455  .63101  -36899  .96354 

51 

1C 

.59484  .63135  .36865   .96349 

50 

11 

.59514  .63170  .36830  .96343 

49 

12 

.59543  .63205   .36795   .96338 

48 

13 

.59573  .63240  .36760  .96333 

47 

14 

.59602  .63275  .36725   .96327 

46 

15 

.59632  .63310  .36690  .96322 

45 

16 

.59661  .63345   .36655   .96316 

44 

17 

.59690  .63379  .36621  .96311 

43 

18 

.59720  .63414  .36586  .96305 

42 

19 

.59749  .63449  .36551   .96300 

41 

2O 

.59778  .63484  .36516  .96294 

40 

21 

.59808  .63519  .36481   .96289 

39 

22 

.59837  .63553  .36447  .96284 

38 

23 

.59866  .63588  .36412  .96278 

37 

24 

.59895   .63623  .36377  .96273 

36 

25 

.59924  .63657  .36343  .96267 

35 

26 

.59954  .63692  .36308  .96262 

34 

27 

.59983  .63726  .36274  .96256 

33 

28 

.60012  .63761  .36239  .96251 

32 

29 

.60041  .63796  .36204  .96245 

31 

30 

.60070  .63830  .36170  .96240 

30 

31 

.60099  .63865   .36135   .96234 

29 

32 

.60128  .63899  .36101   .96229 

28 

33 

.60157  .63934  .36066  .96223 

27 

34 

.60186  .63968  .36032  .96218 

26 

35 

.60215   .64003  .35997  .96212 

25 

36 

.60244  .64037  .35963   .96207 

24 

37 

.60273  .64072  .35928  .96201 

23 

38 

.60302  .64106  .35894  .96196 

22 

39 

.60331  .64140  .35860  .96190 

21 

4O 

.60359  .64175   .35825    .96185 

2O 

41 

.60388  .64209  .35791    .96179 

19 

42 

.60417  .64243   .35757   .96174 

18 

43 

.60446  .64278  .35722   .96168 

17 

44 

.60474  .64312  .35688  .96162 

16 

45 

.60503  .64346  .35654   .96157 

15 

46 

.60532  .64381  .35619   .96151 

14 

47 

.60561  .64415   .35585   .96146 

13 

48 

.60589  .64449  .35551    .96140 

12 

49 

.60618  .64483  .35517   .96135 

11 

50 

.60646  .64517  .35483   .96129 

1O 

51 

.60675   .64552  .35448   .96123 

9 

52 

.60704  .64586  .35414   .96118 

8 

53 

.60732  .64620  .35380   .96112 

7 

54 

.60761  .64654  .35346  .96107 

6 

55 

.60789  .64688  .35312  .96101 

5 

56 

.60818  .64722  .35278  .96095 

4 

57 

.60846  .64756  .35244  .96090 

3 

58 

.60875   .64790  .35210  .96084 

2 

59 

.60903  .64824  .35176  .96079 

1 

60 

.60931  .64858  .35142  .96073 

O 

/ 

9Lcos  91  cot  lOLtan  9Lsin 

/ 

/ 

9Lsin 

9Ltan 

lOLcot 

9Lcos 

/ 

O 

.60931 

.64858 

.35  142 

.96073 

6O 

1 

.60960 

.64892 

.35  108 

.96067 

59 

2 

.60988 

.64  926 

.35  074 

.96062 

58 

3 

.61016 

.64960 

.35  040 

.96056 

57 

4 

.61  045 

.64994 

.35  006 

.96050 

56 

5 

.61  073 

.65  028 

.34972 

.96045 

55 

6 

.61  101 

.65  062 

.34938 

.96039 

54 

7 

.61  129 

.65096 

.34904 

•96034 

53 

8 

.61  158 

.65  130 

.3^870 

.96028 

52 

9 

.61  186 

.65  164 

.34836 

.96022 

51 

1O 

.61  214 

.65  197 

.34803 

.96017 

5O 

11 

.61  242 

.65  231 

.34  769 

.96011 

49 

12 

.61  270 

.65  265 

.34735 

.96005 

48 

13 

.61  298 

.65  299 

.34  701 

.96000 

47 

14 

.61  326 

.65333 

.34667 

.95  994 

46 

15 

.61  354 

.65  366 

.34634 

.95988 

45 

16 

.61  382 

.65  400 

.34600 

.95  982 

44 

17 

.61411 

.65  434 

.34  566 

.95  977 

43 

18 

.61  438 

.65  467 

.34  533 

.95971 

42 

19 

.61  466 

.65  501 

.34499 

.95  965 

41 

2O 

.61  494 

.65  535 

.34465 

.95  960 

40 

21 

.61  522 

.65  568 

.34  432 

.95  954 

39 

22 

.61  550 

.65  602 

.34398 

.95  948 

38 

23 

.61  578 

.65  636 

.34364 

.95  942 

37 

24 

.61  606 

.6T669 

.34331 

.95937 

36 

25 

.61  634 

.65  703 

.34297 

.95931 

35 

26 

.61  662 

.65  736 

.34  264 

.95  925 

34 

27 

.61  689 

.65  770 

.34  230 

.95920 

33 

28 

.61  717 

.65  803 

.34  197 

.95  914 

32 

29 

.61  745 

.65  837 

.34163 

.95  908 

31 

30 

.61  773 

.65  870 

.34  130 

.95  902 

30 

31 

.61  800 

.65  904 

.34096 

.95  897 

29 

32 

.61  828 

.65937 

.34063 

.95  891 

28 

33 

.61  856 

.65  971 

.34  029 

.95  885 

27 

34 

.61  883 

.66004 

.33  996 

.95  879 

26 

35 

.61911 

.66038 

.33  962 

.95  873 

25 

35 

.61  939 

.66071 

.33  929 

.95  868 

24 

37 

.61  966 

.66104 

.33  896 

.95  862 

23 

38 

.61  994 

.66138 

.33862 

.95  856 

22 

39 

.62021 

.66171 

.33  829 

.95  850 

21 

40 

.62049 

.66204 

.33  796 

.95  844 

2O 

41 

.62  076 

.66238 

.33  762 

.95  839 

19 

42 

.62  104 

.66271 

.33  729 

.95  833 

18 

43, 

.62  131 

.66304 

.33  696 

.95  827 

17 

—  ^ 

44 

.62  159 

.66337 

.33663 

.95  821 

16 

45 

.62  186 

.66371 

.33  629 

.95  815 

15 

46 

.62  214 

.66404 

.33  596 

.95  810 

14 

47 

.62  241 

.66437 

.33  563 

.95  804 

13 

48 

.62  268 

.66470 

.33  530 

.95  798 

12 

49 

.62  296 

.66503 

.33497 

.95  792 

11 

5O 

.62323 

.66537 

.33  463 

.95  786 

1O 

51 

.62  350 

.66570 

.33  430 

.95  780 

9 

52 

.62377 

.66603 

.33397 

.95  775 

8 

53 

.62  405 

.66636 

.33  364 

.95  769 

7 

54 

.62432 

.66669 

.33331 

.95  763 

6 

55 

.62  459 

.66  702 

.33  298 

.95  757 

5 

56 

.62  486 

.66  735 

.33  265 

.95  751 

4 

57 

.62513 

.66  768 

.33  232 

.95  745 

3 

58 

.62  541 

.66801 

.33  199 

.95  739 

2 

59 

.62  568 

.66834 

.33  166 

.95  733 

'  1 

60 

.62595 

.66867 

.33  133 

.95  728 

O 

/ 

9Lcos 

9  Loot  10  L  tan 

9  L  sin   / 

66 


65C 


48 


25 


26 


/      DLsin  9Ltan  lOLcot  9Lcos 

/ 

O 

.62595   .66867  .33133  .95728 

6O 

I 

.62622  .66900  .33100  .95722 

59 

2 

.62649  .66933  .33067  .95716 

58 

3 

.62676  .66966  .33034  .95710 

57 

4 

.62703  .66999  .33001  .95704 

56 

5 

.62730  .67032  .32968  .95698 

55 

6 

.62757  .67065   .32935   .95692 

54 

7 

.62784  .67098  .32902  .95686 

53 

8 

.62811   .67131   .32869  .95680 

52 

9 

.62838  .67163  .32837  .95674 

51 

10 

.62865   .67196  .32804  .95668 

50 

11 

.62892  .67229  .32771   .95663 

49 

12 

.62918  .67262  .32738  .95657 

48 

13 

.62945   .67295  .32705   .95651 

47 

14 

.62972  .67327  .32673   .95645 

46 

15 

.62999  .67360  .32640  .95639 

45 

16 

.63026  .67393  .32607  .95633 

44 

17 

.63052  .67426  .32574  .95627 

43 

18 

.63079  .67458  .32542  .95621 

42 

19 

.63106  .67491   .32509  .95615 

41 

2O 

.63133  .67524  .32476  .95609 

40 

21 

.63159  .67556  .32444  .95603 

39 

22 

.63186  .67589  .32411   .95597 

38 

23 

.63213  .67622  .32378  .95591 

37 

24 

.63239  .67654  .32346  .95585 

36 

25 

.63266  .67687  .32313  .95579 

35 

26 

.63292  .67719  .32281   .95573 

34 

27 

.63319  .67752  .32248  .95567 

33 

28 

.63345   .67785  .32215   .95561 

32 

29 

.63372  .67817   .32183  .95555 

31 

30 

.63398  .67850  .32150  .95549 

30 

31 

.63425   .67882  .32118  .95543 

29 

32 

.63451  .67915   .32085  .95537 

28 

33 

.63478  .67947  .32053  .95531 

27 

34 

.63504  .67980  .32020  .95525 

26 

35 

.63531  .68012  .31988  .95519 

25 

36 

.63557  .68044  .31956  .95513 

'24 

37 

.63583  .68077  .31923  .95507 

23 

38 

.63610  .68109  .31891   .95500 

22 

39 

.63636  .68142  .31858  .95494 

21 

40 

.63662  .68174  .31826  .95488 

2O 

41 

.63689  .68206  .31794  .95482 

19 

42 

.63715   .68239  .31761   .95476 

18 

43 

.63741   .68271   .31729  .95470 

17 

44 

.63767   .68303  .31697  .95464 

16 

45 

.63794  .68336  .31664  .95458 

15 

46 

.63820  .68368  .31632  .95452 

14 

47 

.63846  .68400  .31600  .95446 

13 

48 

.63872  .68432  .31568  .95440 

12 

49 

.63898  .68465   .31535   .95434 

11 

5O 

.63924  .68497  .31503   .95427 

10 

51 

.63950  .68529  .31471   .95421 

9 

52 

.63976  .68561   .31439  .95415 

8 

53 

.64002  .68593  .31407  .95409 

7 

54 

.64028  .68626  .31374  .95403 

6 

55 

.64054  .68658  .31342  .95397 

5 

56 

.64080  .68690  .31310  .95391 

4 

57 

.64106  .68722  .31278  .95384 

3 

58 

.64132  .68754  .31246  .95378 

2 

59 

.64158  .68786  .31214  .95372 

1 

6O 

.64184  .68818  .31182  .95366 

O 

/ 

9  L  cos  9  L  cot  10  L  tan  9  L  sin 

/ 

/ 

9Lsin 

9Ltan 

1O  L  cot 

9  L  cos 

/ 

O 

.64  184 

.68818 

.31  182 

.95  366 

6O 

1 

.64  210 

.68850 

.31150 

.95  360 

59 

2 

.64  236 

.68882 

.31118 

.95  354 

58 

3 

.64  262 

.68914 

.31086 

.95  348 

57 

4 

.64  288 

.68946 

.31  054 

.95  341 

56 

5 

.64313 

.68978 

.31022 

.95  335 

55 

6 

.64339 

.69010 

.30990 

.95  329 

54 

7 

.64365 

.69042 

.30958 

.95  323 

53 

8 

.64391 

.69074 

.30926 

.95317 

52 

9 

.64417 

.69  106 

.30894 

.95310 

51 

10 

.64442 

.69  138 

.30862 

.95  304 

5O 

11 

.64  468 

.69  170 

.30830 

.95  298 

49 

12 

.64494 

.69  202 

.30  798 

.95  292 

48 

13 

.64519 

.69  234 

.30  766 

.95  286 

47 

14 

.64545 

.69266 

.30  734 

.95  279 

46 

15 

.64571 

.69  298 

.30  702 

.95  273 

45 

16 

.64  596 

.69329 

.30671 

.95  267 

44 

17 

.64622 

.69361 

.30639 

.95  261 

43 

18 

.64647 

.69393 

.30607 

.95  254 

42 

19 

.64673 

.69425 

.30575 

.95  248 

41 

2O 

.64698 

.69457 

.30  543 

.95  242 

4O 

21 

.64  724 

.69488 

.30512 

.95  236 

39 

22 

.64  749 

.69520 

.30480 

.95  229 

38 

23 

.64  775 

.69552 

.30448 

.95  223 

37 

24 

.64800 

.69  584 

.30416 

.95217 

36 

25 

.64826 

.69615 

.30385 

.95211 

35 

26 

.64851 

.69647 

.30353 

.95  204 

34 

27 

.64877 

.69  679 

.30321 

.95  198 

33 

28 

.64902 

•69710 

.30290 

.95  192 

32 

29 

.64927 

.69  742 

.30258 

.95  185 

31 

3O 

.64953 

.69  774 

.30226 

.95  179 

3O 

31 

.64978 

.69805 

.30  195 

.95  173 

29 

32 

.65  003 

.69837 

.30163 

.95  167 

28 

33 

.65  029 

.69868 

.30  132 

.95  160 

27 

34 

.65  054 

.69900 

.30  100 

.95  154 

26 

35 

.65  079 

.69932 

.30068 

.95  148 

25 

36 

.65  104 

.69963 

.30037 

.95141 

24 

37 

.65  130 

.69995 

.30005 

.95  135 

23 

38 

.65  155 

.70026 

.29974 

.95  129 

22 

39 

.65  180 

.70058 

.29942 

.95  122 

21 

40 

.65  205 

.70089 

.29911 

.95116 

20 

41 

.65  230 

.70121 

.29879 

.95  110 

19 

42 

.65  255 

.70152 

.29  848 

.95  103 

18 

43 

.65  281 

.70184 

.29816 

.95  097 

17 

44 

.65  306 

.70215 

.29  785 

.95  090 

16 

45 

.65331 

.70247 

.29  753 

.95  084 

15 

46 

.65  356 

.70278 

.29  722 

.95  078 

14 

47 

.65  381 

.70309 

.29691 

.95  071 

13 

48 

.65  406 

.70341 

.29  659 

.95  065 

12 

49 

.65431 

.70372 

.29628 

.95  059 

11 

50 

.65  456 

.70404 

.29596 

.95  052 

1O 

51 

.65  481 

.70435 

.29565 

.95  046 

9 

52 

.65  506 

.70466 

.29  534 

.95  039 

8 

53 

.65  531 

.70498 

.29  502 

.95  033 

7 

54 

.65  556 

.70  529 

.29471 

.95  027 

6 

55 

.65  580 

.70560 

.29440 

.95  020 

5 

56 

.65  605 

.70  592 

.29408 

.95  014 

4 

57 

.65  630 

.70623 

.29377 

.95  007 

3 

58 

.65  655 

.70654 

.29346 

.95  001 

2 

59 

.65  680 

.70685 

.29315 

.94995 

1 

60 

.65  705 

.70717 

.29283 

.94988 

O 

/ 

9Lcos 

9Lcot 

10  L  tan 

9Lsin 

/ 

64C 


63C 


27'- 


28C 


49 


/ 

9  L  sin  9  L  tan  10  L  cot  9  L  cos 

/ 

o 

.65705   .70717  .29283  .94988 

00 

1 

.65729  .70748  .29252  .94982 

59 

2 

.65754   .70779  .29221   .94975 

58 

3 

.65779  .70810  .29190  .94969 

57 

4 

.65804  .70841   .29159  .94962 

56 

5 

.65828  .70873  .29127  .94956 

55 

6 

.65853  .70904  .29096  .94949 

54 

7 

.65878  .70935  .29065  .94943 

53 

8 

.65902  .70966  .29034  .94936 

52 

9 

.65927  .70997  .29003  .94930 

51 

1C 

.65952  .71028  .28972  .94923 

50 

11 

.65976  .71059  .28941   .94917 

49 

12 

.66001   .71090  .28910  .94911 

48 

13 

.66025  .71121   .28879  .94904 

47 

14 

.66050  .71153  .28847  .94898 

46 

15 

.66075   .71184  .28816  .94891 

45 

16 

.66099  .71215   .28785   .94885 

44 

17 

.66124  .71246  .28754  .94878 

43 

18 

.66148  .71277  .28723  .94871 

42 

19 

.66173  .71308  .28692  .94865 

41 

20 

.66197  .71339  .28661  .94858 

4O 

21 

.66221   .71370  .28630  .94852 

39 

22 

.66246  .71401   .28599  .94845 

38 

23 

.66270  .71431   .28569  .94839 

37 

24 

.66295   .71462  .28538  .94832 

36 

25 

.66319  .71493  .28507  .94826 

35 

26 

.66343  .71524  .28476  .94819 

34 

27 

.66368  .71555  .28445  .94813 

33 

28 

.66392  .71586  .28414  .94806 

32 

29 

.66416  .71617  .28383  .94799 

31 

SO 

.66441   .71648  .28352  .94793 

3O 

3H.66465   .71679  .28321   .94786 

29 

32 

.66489  .71709  .28291   .94780 

28 

33 

.66513  .71740  .28260  .94773 

27 

34 

.66537  .71771  .28229  .94767 

26 

35 

.66562  .71802  .28198  .94760 

25 

36 

.66586  .71833  .28167  .94753 

24 

37 

.66610  .71863  .28137  .94747 

23 

38 

.66634  .71894  .28106  .94740 

22 

39 

.66658  .71925  .28075  .94734 

21 

4O 

.66682  .71955  .28045   .94727 

2O 

41 

.66706  .71986  .28014  .94720 

19 

42 

.66731  .72017  .27983  .94714 

18 

43 

.66755   .72048  .27952  .94707 

17 

44 

.66779  .72078  .27922  .94700 

16 

45 

.66803   .72109  .27891   .94694 

15 

46 

.66827   .72140  .27860  .94687 

14 

47 

.66851   .72170  .27830  .94680 

13 

48 

.66875   .72201   .27799  .94674 

12 

49 

.66899  .72231  .27769  .94667 

11 

50 

.66922  .72262  .27738  .94660 

1O 

51 

.66946  .72293   .27707  .94654 

9 

52 

.66970  .72323   .27677  .94647 

8 

53 

.66994  .72354  .27646  .94640 

7 

54 

.67018  .72384  .27616  .94634 

6 

55 

.67042   .72415   .27585   .94627 

5 

56 

.67066  .72445   .27555   .94620 

4 

57 

.67090  .72476  .27524  .94614 

3 

58 

.67113   .72506  .27494  .94607 

2 

59 

.67137   .72537  .27463  .94600 

1 

6O 

.67161   .72567  .27433  .94593 

O 

/ 

9Lcos  9Lcot  10  L  tan  9Lsin 

/ 

/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

0 

.67161  .72567  .27433  .94593 

60 

1 

.67185  .72598  .27402  .94587 

59 

2 

.67208  .72628  .27372  .94580 

58 

3 

.67232  .72659  .27341  .94573 

57 

4 

.67256  .72689  .27311  .94567 

56 

5 

.67280  .72720  .27280  .'94560 

55 

6 

.67303   .72750  .27250  .94553 

54 

7 

.67327  .72780  .27220  .94546 

53 

8 

.67350  .72811   .27189  .94540 

52 

9 

.67374  .72841   .27159  .94533 

51 

1O 

.67398  .72872  .27128  .94526 

5O 

11 

.67421  .72902  .27098  .94519 

49 

12 

.67445   .72932  .27068  .94513 

48 

13 

.67468  .72963  .27037  .94506 

47 

14 

.67492  .72993  .27007  .94499 

46 

15 

.67515  .73023  .26977  .94492 

45 

16 

.67539  .73054  .26946  .94485 

44 

17 

.67562  .73084  .26916  .94479 

43 

18 

.67586  .73114  .26886  .94472 

42 

19 

.67609  .73144  .26856  .94465 

41 

2O 

.67633  .73175   .26825   .94458 

4O 

21 

.67656  .73205   .26795   .94451 

39 

22 

.67680  .73235  .26765   .94445 

38 

23 

.67703  .73265  .26735   .94438 

37 

24 

.67726  .73295   .26705   .94431 

36 

25 

.67750  .73326  .26674  .94424 

35 

26 

.67773  .73356  .26644  .94417 

34 

27 

.67796  .73386  .26614  .94410 

33 

28 

.67820  .73416  .26584  .94404 

32 

29 

.67843  .73446  .26554  .94397 

31 

3O 

.67866  .73476  .26524  .94390 

30 

31 

.67890  .73507  .26493   .94383 

29 

32 

.67913  .73537  .26463  .94376 

28 

33 

.67936  .73567  .26433  .94369 

27 

34 

-.67959  .73597  .26403  .94362 

26 

35 

.67982  .73627  .26373  .94355 

25 

36 

.68006  .73657  .26343  .94349 

24 

37 

.68029  .73687  .26313   .94342 

23 

38 

.68052  .73717  .26283   .94335 

22 

39 

.68075   .73747  .26253  .94328 

21 

40 

.68098  .73777  .26223  .94321 

2O 

41 

.68121   .73807  .26193  .94314 

19 

42 

.68144  .73837  .26163   .94307 

18 

43 

.68167  .73867  .26133  .94300 

17 

44 

.68190  .73897  .26103  .94293 

16 

45 

.68213  .73927  .26073  .94286 

15 

46 

.63237  .73957   .26043   .94279 

14 

47 

.68260  .73987   .26013  .94273 

13 

48 

.68283  .74017  .25983   .94266 

12 

49 

.68305  .74047  .25953  .94259 

11 

5O 

.68328  .74077  .25923   .94252 

10 

51 

.68351   .74107  .25893   .94245 

9 

52 

.68374  .74137  .25863   .94238 

8 

53 

.68397  .74166  .25834  .94231 

7 

54 

.68420  .74196  .25804  .94224 

6 

55 

.68443  .74226  .25774  .94217 

5 

56 

.68466  .74256  .25744  .94210 

4 

57 

.68489  .74286  .25714  .94203 

3 

58 

.68512  .74316  .25684  .94196 

2 

59 

.68534  .74345   .25655   .94189 

1 

6O 

.68557  .74375   .25625   .94182 

O 

/ 

9Lcos  9Lcot  lOLtan  9Lsin 

/ 

62' 


61 


50 


29 


30< 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

0 

.68557  .74375  .25625  .94182 

60 

1 

.68580  .74405  .25595  .94175 

59 

2 

.68603  .74435  .25565  .94168 

58 

3 

.68625  .74465  .25535  .94161 

57 

4 

.68648  .74494  .25506  .94154 

56 

5 

.68671  .74524  .25476  .94147 

55 

6 

.68694  .74554  .25446  .94140 

54 

7 

.68716  .74583  .25417  .94133 

53 

8 

.68739  .74613  .25387  .94126 

52 

9 

.68762  .74643  .25357  .94119 

51 

10 

.68784  .74673  .25327  .94112 

50 

11 

.68807  .74702  .25298  .94105 

49 

12 

.68829  .74732  .25268  .94098 

48 

13 

.68852  .74762  .25238  .94090 

47 

14 

.68875  .74791  .25209  .94083 

46 

15 

.68897  .74821  .25179  .94076 

45 

16 

.68920  .74851  .25149  .94069 

44 

17 

.68942  .74880  .25120  .94062 

43 

18 

.68965  .74910  .25090  .94055 

42 

19 

.68987  .74939  .25061  .94048 

41 

20 

.69010  .74969  .25031  .94041 

4O 

21 

.69032  .74998  .25002  .94034 

39 

22 

.69055  .75028  .24972  .94027 

38 

23 

.69077  .75058  .24942  .94020 

37 

24 

.69100  .75087  .24913  .94012 

36 

25 

.69122  .75117  .24883  .94005 

35 

26 

.69144  .75146  .24854  .93998 

34 

27 

.69167  .75176  .24824  .93991 

33 

28 

.69189  .75205  .24795  .93984 

32 

29 

.69212  .75235  .24765  .93977 

31 

30 

.69234  .75264  .24736  .93970 

30 

31 

.69256  .75294  .24706  .93963 

29 

32 

.69279  .75323  .24677  .93955 

28 

33 

.69301  .75353  .24647  .93948 

27 

34 

.69323  .75382  .24618  .93941 

26 

35 

.69345  .75411  .24589  .93934 

25 

36 

.69368  .75441  .24559  .93927 

24 

37 

.69390  .75470  .24530  .93920 

23 

38 

.69412  .75500  .24500  .93912 

22 

39 

.69434  .75529  .24471  .93905 

21 

40 

.69456  .75558  .24442  .93898 

20 

41 

.69479  .75588  .24412  .93891 

19 

42 

.69501  .75617  .24383  .93884 

18 

43 

.69523  .75647  .24353  .93876 

17 

44 

.69545  .75676  .24324  .93869 

16 

45 

.69567  .75705  .24295  .93862 

15 

46 

.69589  .75735  .24265  .93855 

14 

47 

.69611  .75764  .24236  .93847 

13 

48 

.69633  .75793  .24207  .93840 

12 

49 

.69655  .75822  .24178  .93833 

11 

5O 

.69677  .75852  .24148  .93826 

1O 

51 

.69699  .75881  .24119  .93819 

9 

52 

.69721  .75910  .24090  .93811 

8 

53 

.69743  .75939  .24061  .93804 

7 

54 

.69765  .75969  .24031  .93797 

6 

55 

.69787  .75998  .24002  .93789 

5 

56 

.69809  .76027  .23973  .93782 

4 

57 

.69831  .76056  .23944  .93775 

3 

58 

.69853  .76086  .23914  .93768 

2 

59 

.69875  .76115  .23885  .93760 

1 

6O 

'.69897  .76144  .23856  .93753 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

O 

.69897  .76144  .23856  .93753 

6O 

1 

.69919  .76173  .23827  .93746 

59 

2 

.69941   .76202  .23798  .93738 

58 

3 

.69963  .76231   .23769  .93731 

57 

4 

.69984  .76261   .23739  .93724 

56 

5 

.70006  .76290  .23710  .93717 

55 

6 

.70028  .76319  .23681    .93709 

54 

7 

.70050  .76348  .23652   .93702 

53 

8 

.70072  .76377  .23623   .93695 

52 

9 

.70093  .76406  .23594  .93687 

51 

1O 

.70115   .76435   .23565   .93680 

5O 

11 

.70137  .76464  .23536  .93673 

49 

12 

.70159  .76493  .23507  .93665 

48 

13 

.70180  .76522  .23478  .93658 

47 

14 

.70202  .76551   .23449  .93650 

46 

15 

.70224  .76580  .23420  .93643 

45 

16 

.70245  .76609  .23391   .93636 

44 

17 

.70267  .76639  .23361   .93628 

43 

18 

.70288  .76668  .23332  .93621 

42 

19 

.70310  .76697  .23303   .93614 

41 

20 

.70332  .76725   .23275   .93606 

40 

21 

.70353  .76754  .23246  .93599 

39 

22 

.70375   .76783  .23217  .93591 

36 

23 

.70396  .76812  .23188  .93584 

37 

24 

.70418  .76841   .23159  .93577 

36 

25 

.70439  .76870  .23130  .93569 

35 

26 

.70461   .76899  .23101  .93562 

34 

27 

.70482  .76928  .23072  .93554 

33 

28 

.70504  .76957  .23043  .93547 

32 

29 

.70525  .76986  .23014  .93539 

31 

30 

.70547  .77015  .22985  .93532 

30 

31 

.70568  .77044  .22956  .93525 

29 

32 

.70590  .77073  .22927  .93517 

28 

33 

.70611   .77101   .22899  .93510 

27 

34 

.70633  .77130  .22870  .93502 

26 

35 

.70654  .77159  .22841   .93495 

25 

36 

.70675   .77188  .22812  .93487 

24 

37 

.70697  .77217  .22783  .93480 

23 

38 

.70718  .77246  .22754  .93472 

22 

39 

.70739  .77274  .22726  .93465 

21 

4O 

.70761   .77303  .22697  .93457 

2O 

41 

.70782  .77332  .22668  .93450 

19 

42 

.70803  .77361   .22639  .93442 

18 

43 

.70824  .77390  .22610  .93435 

17 

44 

.70846  .77418  .22582  .93427 

16 

45 

.70867  .77447  .22553  .93420 

15 

46 

.70888  .77476  .22524  .93412 

14 

47 

.70909  .77505   .22495  .93405 

13 

48 

.70931   .77533  .22467  .93397 

12 

49 

.70952  .77562  .22438  .93390 

11 

50 

.70973  .77591  .22409  .93382 

1O 

51 

.70994  .77619  .22381  .93375 

9 

52 

.71015   .77648  .22352  .93367 

8 

53 

.71036  .77677  .22323  .93360 

7 

54 

.71058  .77706  .22294  .93352 

6 

55 

.71079  .77734  .22266  .93344 

5 

56 

.71100  .77763   .22237  .93337 

4 

57 

.71121  .77791   .22209  .93329 

3 

58 

.71142  .77820  .22180  .93322 

2 

59 

.71163  .77849  .22151  .93314 

1 

60 

.71184  .77877  .22123  .93307 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

60 


59 


32' 


51 


/ 

9Lsin  9Ltan  lOLcot9Lcos 

/ 

o 

.71184  .77877  .22123  .93307 

7*0^ 

1 

.71205   .77906  .22094  .93299 

59 

2 

.71226  .77935   .22065  .93291 

58 

3 

.71247  .77963   .22037  .93284 

57 

4 

.71268  .77992  .22008  .93276 

56 

5 

.71289  .78020  .21980  .93269 

55 

6 

.71310  .78049  .21951  .93261 

54 

7 

.71331  .78077  .21923  .93253 

53 

8 

.71352  .78106  .21894  .93246 

52 

9 

.71373   .78135   .21865  .93238 

51 

1O 

.71393  .78163  .21837  .93230 

5O 

11 

.71414  .78192  .21808  .93223 

49 

12 

.71435  .78220  .21780  .93215 

48 

13 

.71456  .78249  .21751    .93207 

47 

14 

.71477  .78277  .21723  .93200 

46 

15 

.71498  .78306  .21694  .93192 

45 

16 

.71519  .78334  .21666  .93184 

44 

17 

.71539  .78363  .21637  .93177 

43 

18 

.71560  .78391  .21609  .93169 

42 

19 

.71581  .78419  .21581  .93161 

41 

20 

.71602  .78448  .21552  .93154 

40 

21 

.71622  .78476  .21524  .93146 

39 

22 

.71643  .78505   .21495   .93138 

38 

23 

.71664  .78533  .21467  .93131 

37 

24 

.71685   .78562  .21438  .93123 

36 

25 

.71705  .78590  .21410  .93115 

35 

26 

.71726  .78618  .21382  .93108 

34 

27 

.71747  .78647  .21353  .93100 

33 

28 

.71767  .78675  .21325  .93092 

32 

29 

.71788  .78704  .21296  .93084 

31 

3O 

.71809  .78732  .21268  .93077 

3O 

31 

.71829  .78760  .21240  .93069 

29 

32 

.71850  .78789  .21211  .93061 

28 

33 

.71870  .78817  .21183  .93053 

27 

34 

.71891   .78845   .21155  .93046 

26 

35 

.71911  .78874  .21126  .93038 

25 

36 

.71932  .78902  .21098  .93030 

24 

37 

.71952  .78930  .21070  .93022 

23 

38 

.71973  .78959  .21041   .93014 

22 

39 

.71994  .78987  .21013   .93007 

21 

40 

.72014  .79015  .20985  .92999 

2O 

41 

.72034  .79043   .20957  .92991 

19 

42 

.72055   .79072  .20928  .92983 

18 

43 

.72075   .79100  .20900  .92976 

17 

44 

.72096  .79128  .20872  .92968 

16 

45 

.72116  .79156  .20844  .92960 

15 

46 

.72137  .79185   .20815   .92952 

14 

47 

.72157  .79213  .20787  .92944 

13 

48 

.72177  .79241  .20759  .92936 

12 

49 

.72198  .79269  .20731   .92929 

11 

50 

.72218  .79297  .20703  .92921 

1O 

51 

.72238  .79326  .20674  .92913 

9 

52 

.72259  .79354  .20646  .92905 

8 

53 

.72279  .79382  .20618  .92897 

7 

54 

.72299  .79410  .20590  .92889 

6 

55 

.72320  .79438  .20562  .92881 

5 

56 

.72340  .79466  .20534  .92874 

4 

57 

.72360  .79495   .20505   .92866 

3 

58 

.72381  .79523  .20477  .92858 

2 

59 

.72401  .79551  .20449  .92850 

1 

60 

.72421  .79579  .20421  .92842 

0 

/ 

9Lcos  9Lcot  lOLtan  9Lsin 

/ 

/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

O 

.72421  .79579  .20421  .92842 

6O 

1 

.72441  .79607  .20393  .92834 

59 

2 

.72461  .79635   .20365  .92826 

58 

3 

.72482  .79663  .20337  .92818 

57 

4 

.72502  .79691  .20309  .92810 

56 

5 

.72522  .79719  .20281  .92803 

55 

6 

.72542  .79747  .20253  .92795 

54 

7 

.72562  .79776  .20224  .92787 

53 

8 

.72582  .79804  .20196  .92779 

52 

9 

.72602  .79832  .20168  .92771 

51 

10 

.72  622  •  .79  860  .20  140  ,.92  763 

5O 

11 

.72643  .79888  .20112  .92755 

49 

12 

.72663   .79916  .20084  .92747 

48 

13 

.72683  .79944  .20056  .92739 

47 

14 

.72703  .79972  .20028  .92731 

46 

15 

.72723  .80000  .20000  .92723 

45 

16 

.72743  .80028  .19972  .92715 

44 

17 

.72763  .80056  .19944  .92707 

43 

18 

.72783  .80084  .19916  .92699 

42 

19 

.72803  .80112  .19888  .92691 

41 

2O 

.72823  .80140  .19860  .92683 

40 

21 

.72843  .80168  .19832  .92675 

39 

22 

.72863  .80195   .19805  .92667 

38 

23 

.72883  .80223  .19777  .92659 

37 

24 

.72902  .80251  .19749  .92651 

36 

25 

.72922  .80279  .19721  .92643 

35 

26 

.72942  .80307  .19693  .92635 

34 

27 

.72962  .80335  .19665   .92627 

33 

28 

.72982  .80363  .19637  .92619 

32 

29 

.73002  .80391  .19609  .92611 

31 

3O 

.73022  .80419  .19581   .92603 

3O 

31 

.73041   .80447  .19553  .92595 

29 

32 

.73061  .80474  .19526  .92587 

28 

33 

.73081  .80502  .19498  .92579 

27 

34 

.73101  .80530  .19470  .92571 

26 

35 

.73121  .80558  .19442  .92563 

25 

36 

.73140  .80586  .19414  .92555 

24 

37 

.73160  .80614  .19386  .92546 

23 

38 

.73180  .80642  .19358  .92538 

22 

39 

.73200  .80669  .19331   .92530 

21 

4O 

.73219  .80697  .19303  .92522 

2O 

41 

.73239  .80725  .19275  .92514 

19 

42 

.73259  .80753  .19247  .92506 

18 

43 

.73278  .80781   .19219  .92498 

17 

44 

.73298  .80808  .19192  .92490 

16 

45 

.73318  .80836  .19164  .92482 

15 

46 

.73337  .80864  .19136  .92473 

14 

47 

.73357  .80892  .19108  .92465 

13 

48 

.73377  .80919  .19081   .92457 

12 

49 

.73396  .80947  .19053   .92449 

11 

5O 

.73416  .80975   .19025  .92441 

1O 

51 

.73435  .81003  .18997  .92433 

9 

52 

.73455   .81030  .18970  .92425 

8 

53 

.73474  .81058  .18942  .92416 

7 

54 

.73494  .81086  .18914  .92408 

6 

55 

.73513  .81113  .18887  .92400 

5 

56 

.73533  .81141  .18859  .92392 

4 

57 

.73552  .81169  .18831   .92384 

3 

58 

.73572  .81196  .18804  .92376 

2 

59 

.73591   .81224  .18776  .92367 

1 

60 

.73611  .81252  .18748  .92359 

O 

/ 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

58C 


57C 


52 


33 


34' 


/ 

9Lsin  9Ltan  lOLcot  9Lcos 

/ 

o 

.73611  .81252  .18748  .92359 

60 

1 

.73630  .81279  .18721  .92351 

59 

2 

.73650  .81307  .18693  .92313 

58 

3 

.73669  .81335  .18665   .92335 

57 

4 

.73689  .81362  .18638  .92326 

56 

5 

.73708  .81390  .18610  .92318 

55 

6 

.73727  .81418  .18582   .92310 

54 

7 

.73747  .81445  .18555   .92302 

53 

8 

.73766  .81473  .18527  .92293 

52 

9 

.73785  .81500  .18500  .92285 

51 

10 

.73805  .81528  .18472  .92277 

5O 

11 

.73824  .81556  .18444  .92269 

49 

12 

.73843  .81583  .18417  .92260 

48 

13 

.73863  .81611  .18389  .92252 

47 

14 

.73882  .81638  .18362  .92244 

46 

15 

.73901   .81666  .18334  .92235 

45 

16 

.73921  .81693  .18307  .92227 

44 

17 

.73940  .81721   .18279  .92219 

43 

18 

.73959  .81748  .18252  .92211 

42 

19 

.73978  .81776  .18224  .92202 

41 

20 

.73997  .81803  .18197  .92194 

4O 

21 

.74017  .81831   .18169  .92186 

39 

22 

.74036  .81858  .18142  .92177 

38 

23 

.74055  .81886  .18114  .92169 

37 

24 

.74074  .81913  .18087  .92161 

36 

25 

.74093  .81941   .18059  .92152 

35 

26 

.74113  .81968  .18032  .92144 

34 

27 

.74132  .81996  .18004  .92136 

33 

28 

.74151  .82023  .17977  .92127 

32 

29 

.74170  .82051  .17949  .92119 

31 

3O 

.74189  .82078  .17922  .92111 

30 

31 

.74208  .82106  .17894  .92102 

29 

32 

.74227  .82133  .17867  .92094 

28 

33 

.74246  .82161   .17839  .92086 

27 

34 

.74265   .82188  .17812  .92077 

26 

35 

.74284  .82215  .17785  .92069 

25 

36 

.74303  .82243  .17757  .92060 

24 

37 

.74322  .82270  .17730  .92052 

23 

38 

.74341  .82298  .17702  .92044 

22 

39 

.74360  .82325  .17675  .92035 

21 

4O 

.74379  .82352  .17648  .92027 

20 

41 

.74398  .82380  .17620  .92018 

19 

42 

.74417  .82407  .17593  .92010 

18 

43 

.74436  .82435  .17565  .92002 

17 

44 

.74455  .82462  .17538  .91993 

16 

45 

.74474  .82489  .17511  .91985 

15 

46 

.74493   .82517  .17483  .91976 

14 

47 

.74512  .82544  .17456  .91968 

13 

48 

.74531  .82571  .17429  .91959 

12 

49 

.74549  .82599  .17401  .91951 

11 

5O 

.74568  .82626  .17374  .91942 

1O 

51 

.74587  .82653  .17347  .91934 

9 

52 

.74606  .82681   .17319  .91925 

8 

53 

.74625  .82708  .17292  .91917 

7 

54 

.74644  .82735  .17265   .91908 

6| 

55 

.74662  .82762  .17238  .91900 

5 

56 

.74681  .82790  .17210  .91891 

4 

57 

.74700  .82817  .17183  .91883 

3 

58 

.74719  .82844  .17156  .91874 

2 

59 

.74737  .82871   .17129  .91866 

1 

60 

.74756  .82899  .17101  .91857 

O 

/ 

9Lcos  9LcotlOLtan9Lsin 

/ 

/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

O 

.74756  .82899  .17101   .91857' 

60 

1 

.74775  .82926  .17074  .91849 

59 

2 

.74794  .82953   .17047   .91840 

58 

3 

.74812  .82980  .17020   .91832 

57 

4 

.74831  .83008  .16992   .91823 

56 

5 

.74850  .83035    .16965   .91815 

55 

6 

.74868  .83062  .16938  .91806 

54 

7 

.74887   .83089  .16911   .91798 

53 

8 

.74906  .83117   .16883   .91789 

52 

9 

.74924   .83144   .16856  .91781 

51 

1O 

.749-13   .83171   .16829  .91772 

50 

11 

.74961   .83198  .16802  .91763 

49 

12 

.74980   .83225    .16775   .91755 

48 

13 

.74999   .83252   .16748  .91746 

47 

14 

.75017   .83280  .16720  .91738 

46 

15 

.75036  .83307  .16693  .91729 

45 

16 

.75054  .83334  .16666  .91720 

44 

17 

.75073  .83361   .16639  .91712 

43 

18 

.75091   .83388  .16612  .91703 

42 

19 

.75110  .83415   .16585   .91695 

41 

2O 

.75128  .83442  .16558  .91686 

4O 

21 

.75  147  .83470  .16530  .91677 

39 

22 

.75165   .83497  .16503  .91669 

38 

23 

.75184  .83524.16476  .91660 

37 

24 

.75202  .83551   .16449  .91651 

36 

25 

.75221   .83578  .16422  .91643 

35 

26 

.75239  .83605   .16395   .91634 

34 

27 

.75258  .83632  .16368  .91625 

33 

28 

.75276  .83659  .16341   .91617 

32 

29 

.7-5294  .83686  .16314  .91608 

31 

30 

.75313   .83713   .16287  .91599 

30 

31 

.75331   .83740  .16260  .91591 

29 

32 

.75350  .83768  .16232  .91582 

28 

33 

.75368  .83795   .16205   .91573 

27 

34 

.75386  .83822  .16178  .91565 

26 

35 

.75405   .83849  .16151   .91556 

25 

36 

.75423   .83876  .16124  .91547 

24 

37 

.75441   .83903  .16097  .91538 

23 

38 

.75459  .83930  .16070  .91530 

22 

39 

.75478  .83957  .16043  .91521 

21 

4O 

.75496  .83984  .16016  .91512 

2O 

41 

.75514  .84011  .15989  .91504 

19 

42 

.75533   .84038  .15962  .91495 

18 

43 

.75551    .84065   .15935   .91486 

17 

44 

.75569  .84092  .15908  .91477 

16 

45 

.75587  .84119  .15881  .91469 

15 

46 

.75605   .84146  .15854  .91460 

14 

47 

.75624  .84173   .15827  .91451 

13 

48 

.75642   .84200   .15800  .91442 

12 

49 

.75660  .84227   .15773  .91433 

11 

50 

.75678  .84254  .15746  .91425 

1OI 

51 

.75696  .84280  .15720  .91416 

9 

52 

.75714   .84307  .15693   .91407 

8 

53 

.75733   .84334  .15666  .91398 

7 

54 

.75751   .84361   .15639  .91389 

6 

55 

.75769  .84388  .15612  .91381 

5 

56 

.75787  .84415   .15585   .91372 

4 

57 

.75805   .84442  .15558  .91363 

3 

58 

.75823   .84469  .15531   .91354 

2 

59 

.75841   .84496  .15504  .91345 

1 

60 

.75859  .84523  .15477  .91336 

O 

~7 

9  L  cos  9  L  cot  1O  L  tan  9  L  sin 

/ 

56 


55 


35 


36< 


53 


/ 

9Lsin 

9Ltan 

1O  L  cot 

9  Lcos 

/ 

0 

.75  859 

.84  523 

.15477 

.91  336 

6O 

1 

.75  877 

.84550 

.15450 

.91  328 

59 

2 

.75  895 

.84  576 

.15  424 

.91319 

58 

3 

.75913 

.84  603 

.15397 

.91  310 

57 

4 

.75931 

.84630 

.15370 

.91  301 

56 

5 

.75949 

.84657 

.15343 

.91  292 

55 

6 

.75  967 

.84684 

.15316 

.91  283 

54 

7 

.75985 

.84711 

.15  289 

.91  274 

53 

8 

.76003 

.84  738 

.15262 

.91  266 

52 

9 

.76021 

.84764 

.15  236 

.91  257 

51 

10 

.76039 

.84  791 

.15  209 

.91  248 

50 

11 

.76057 

.84818 

.15  182 

.91  239 

49 

12 

.76075 

.84  845 

.15  155 

.91  230 

48 

13 

.76093 

.84  872 

.15  128 

.91  221 

47 

14 

.76111 

.84  899 

.15101 

.91212 

46 

15 

.76  129 

.84925 

.15075 

.91  203 

45 

16 

.76  146 

.84952 

.15048 

.91  194 

44 

17 

.76  164 

.84979 

.15021 

.91  185 

43 

18 

.76  182 

.85  006 

.14994 

.91  176 

42 

19 

.76  200 

.85033 

.14967 

.91  167 

41 

2O 

.76218 

.85  059 

.14941 

.91  158 

40 

21 

.76  236 

.85  086 

.14914 

.91  149 

39 

22 

.76253 

.85  113 

.14887 

.91  141 

38 

23 

.76271 

.85  140 

.14860 

.91  132 

37 

24 

.76  289 

.85  166 

.14834 

.91  123 

36 

25 

.76307 

.85  193 

.14807 

.91114 

35 

26 

.76324 

.85  220 

.14780 

.91  105 

34 

27 

.76342 

.85  247 

.14753 

.91  096 

33 

28 

.76360 

.85  273 

.14  727 

.91087 

32 

29 

.76378 

.85  300 

.14  700 

.91  078 

31 

30 

.76395 

.85  327 

.14673 

.91  069 

30 

31 

.76413 

.85  354 

.14646 

.91  060 

29 

32 

.76431 

.85  380 

.14620 

.91051 

28 

33 

.76448 

.85  407 

.14593 

.91  042 

27 

34 

.76466 

.85  434 

.14  566 

.91033 

26 

35 

.76484 

.85  460 

.14540 

.91  023 

25 

36 

.76501 

.85  487 

.14513 

.91014 

24 

37 

.76519 

.85  514 

.14486 

.91  005 

23 

38 

.76537 

.85  540 

.14460 

.90996 

22 

39 

.76554 

.85  567 

.14433 

.90987 

21 

4O 

.76572 

.85  594 

.14406 

.90978 

2O 

41 

.76  590 

.85  620 

.14380 

.90969 

19 

42 

.76607 

.85  647 

.14353 

.90960 

18 

43 

.76625 

.85  674 

.14326 

.90951 

17 

44 

.76642 

.85  700 

.14300 

.90942 

16 

45 

.76660 

.85  727 

.14273 

.90933 

15 

46 

.76677 

.85  754 

.14246 

.90924 

14 

47 

.76695 

.85  780 

.14220 

.90915 

13 

48 

.76712 

.85  807 

.14  193 

.90906 

12 

49 

.76  730 

.85  834 

.14  166 

.90896 

11 

50 

.76  747 

.85  860 

.14  140 

.90887 

1O 

51 

.76  765 

.85  887 

.14113 

.90878 

9 

52 

.76  782 

.85  913 

.14087 

.90869 

8 

53 

.76800 

.85  940 

.14060 

.90860 

7 

54 

.76817 

.85  967 

.14033 

.90851 

6 

55 

.76835 

.85  993 

.14007 

.90842 

5 

56 

.76852 

.86020 

.13  980 

.90832 

4 

57 

.76870 

.86046 

.13954 

.90823 

3 

58 

.76887 

.86073 

.13927 

.90814 

2 

59 

.76904 

.86  103 

.13900 

.90805 

1 

60 

.76922 

.86  126 

.13874 

.90  796 

O 

/ 

9  Lcos 

9Lcot 

10  L  tan 

9Lsin 

/ 

/ 

91  sin 

9Ltan 

lOLcot 

9  Lcos 

t 

O 

.76922 

.86126 

.13874 

.90  796 

60 

1 

.76939 

.86153 

.13847 

.90787 

59 

2 

.76957 

.86179 

.13821 

.90777 

58 

3 

.76974 

.86  206 

.13  794 

.90  768 

57 

4 

.76991 

.86232 

.13  768 

.90759 

56 

5 

.77009 

.86259 

.13  741 

.90  750 

55 

6 

.77026 

.86285 

.13  715 

.90  741 

54 

7 

.77043 

.86312 

.13688 

.90  731 

53 

8 

.77061 

.86338 

.13662 

.90  722 

52 

9 

.77078 

.86365 

.13635 

.90713 

51 

1O 

.77  095 

.86392 

.13608 

.90  704 

5O 

11 

.77112 

.86418 

.13  582 

.90694 

49 

12 

.77  130 

.86445 

.13  555 

.90685 

48 

13 

.77  147 

.86471 

.13  529 

.90676 

47 

14 

.77  164 

.86498 

.13  502 

.90667 

46 

15 

.77181 

.86524 

.13476 

.90657 

45 

16 

.77  199 

.86551 

.13449 

.90648 

44 

17 

.77216 

.86577 

.13  423 

.90639 

43 

18 

.77233 

.86603 

.13397 

.90630 

42 

19 

.77250 

.86630 

.13370 

.90620 

41 

2O 

.77  268 

.86656 

.13  344 

.90611 

40 

21 

.77285 

.86683 

.13317 

.90602 

39 

22 

.77302 

.86  709 

.13291 

.90592 

38 

23 

.77319 

.86  736 

.13  264 

.90  583 

37 

24 

.77336 

.86762 

.13  238 

.90574 

36 

25 

.77353 

.86  789 

.13211 

.90565 

35 

26 

.77370 

.86815 

.13  185 

.90555 

34 

27 

.77387 

.86842 

.13158 

.90  546 

33 

28 

.77405 

.86868 

.13  132 

.90537 

32 

29 

.77422 

.86894 

.13  106 

.90527 

31 

30 

.77439 

.86921 

.13079 

.90518 

30 

31 

.77456 

.86947 

.13053 

.90  509 

29 

32 

.77473 

.86974 

.13026 

.90499 

28 

33 

.77  490 

.87000 

.13000 

.90490 

27 

34 

.77507 

.87027 

.12973 

.90480 

26 

35 

.77524 

.87053 

.12947 

.90471 

25 

36 

.77541 

.87079 

.12921 

.90462 

24 

37 

.77558 

.87  106 

.12  894 

.90452 

23 

38 

.77575 

.87  132 

.12868 

.90443 

22 

39 

.77  592 

.87  158 

.12842 

.90434 

21 

4O 

.77609 

.87  185 

.12815 

.90424 

2O 

41 

.77  626 

.87211 

.12  789 

.90415 

19 

42 

.77  643 

.87  238 

.12  762 

.90405 

18 

43 

.77660 

.87264 

.12  736 

.90396 

17 

44 

.77677 

.87  290 

.12710 

.90386 

16 

45 

.77694 

.87317 

.12683 

.90377 

15 

46 

.77711 

.87343 

.12657 

.90368 

14 

47 

.77  728 

.87  369 

.12631 

.90358 

13 

48 

.77  744 

.87  396 

.12604 

.90349 

12 

49 

.77761 

.87422 

.12578 

.90339 

11 

5O 

.77  778 

.87448 

.12552 

.90330 

10 

51 

.77  795 

.87475 

.12525 

.90320 

9 

52 

.77812 

.87501 

.12499 

.90311 

8 

53 

.77829 

.87  527 

.12473 

.90301 

7 

54 

.77  846 

.87554 

.12  446 

.90292 

6 

55 

.77  862 

.87  580 

.12420 

.90282 

5 

56 

.77879 

.87  606 

.12394 

.90273 

4 

57 

.77896 

.87  633 

.12367 

.90263 

3 

58 

.77913 

.87659 

.12341 

.90254 

2 

59 

.77930 

.87685 

.12315 

.90244 

1 

60 

.77946 

.87711 

.12289 

.90235 

O 

/ 

9  Lcos 

9Lcot 

10  L  tan 

9Lsin 

/ 

54C 


53' 


54 


37° 


38 


/ 

9Lsin 

9Ltan 

lOLcot 

9  Lcos 

/ 

o 

.77946 

.87711 

.12289 

.90  235 

60 

1 

.77963 

.87  738 

.12262 

.90  225 

59 

2 

.77980 

.87  764 

.12236 

.90216 

58 

3 

.77997 

.87  790 

.12210 

.90  206 

57 

4 

.78013 

.87817 

.12183 

.90  197 

56 

5 

.78030 

.87  843 

.12157 

.90187 

55 

6 

.78047 

.87  869 

.12131 

.90178 

54 

7 

.78063 

.87895 

.12105 

.90  168 

53 

8 

.78080 

.87922 

.12078 

.90  159 

52 

9 

.78097 

.87948 

.12052 

.90  149 

51 

1O 

.78113 

.87974 

.12026 

.90  139 

50 

11 

.78  130 

.88000 

.12000 

.90  130 

49 

12 

.78  147 

.88027 

.11973 

.90  120 

48 

13 

.78  163 

.88053 

.11947 

.90111 

47 

14 

.78  180 

.88079 

.11921 

.90  101 

46 

15 

.78  197 

.88  105 

.11895 

.90091 

45 

16 

.78213 

.88131 

.11869 

.90082 

44 

17 

.78  230 

.88  158 

.11  842 

.90072 

43 

18 

.78  246 

.88  184 

.11816 

.90063 

42 

19 

.78  263 

.88210 

.11790 

.90053 

41 

20 

.78280 

.88236 

.11764 

.90043 

40 

21 

.78  296 

.88  262 

.11738 

.90034 

39 

22 

.78313 

.88  289 

.11711 

.90024 

38 

23 

.78329 

.88315 

.11685 

.90014 

37 

24 

.78346 

.88341 

.11659 

.90005 

36 

25 

.78362 

.88367 

.11633 

.89995 

35 

26 

.78379 

.88393 

.11607 

.89985 

34 

27 

.78  395 

.88420 

.11580 

.89976 

33 

28 

.78412 

.88446 

.11554 

.89966 

32 

29 

.78428 

.88472 

.11528 

.89956 

31 

30 

.78445 

.88498 

.11502 

.89947 

3O 

31 

.78461 

.88  524 

.11476 

.89937 

29 

32 

.78478 

.88550 

.11450 

.89927 

28 

33 

.78494 

.88577 

.11423 

.89918 

27 

34 

.78510 

.88603 

.11397 

.89908 

26 

35 

.78527 

.88629 

.11371 

.89898 

25 

36 

.78543 

.88655 

.11345 

.89888 

24 

37 

.78560 

.88681 

.11319 

.89879 

23 

38 

.78576 

.88  707 

.11293 

.89869 

22 

39 

.78  592 

.88  733 

.11267 

.89859 

21 

40 

.78609 

.88759 

.11241 

.89849 

2O 

41 

.78625 

.88  786 

.11214 

.89840 

19 

42 

.78642 

.88812 

.11188 

.89830 

18 

43 

.78658 

.88838 

.11162 

.89  820 

17 

44 

.78674 

.88864 

.11136 

.89810 

16 

45 

.78691 

.88890 

.11110 

.89801 

15 

46 

.78  707 

.88916 

.11084 

.89791 

14 

47 

.78  723 

.88942 

.11058 

.89781 

13 

48 

.78739 

.88968 

.11032 

.89771 

12 

49 

.78756 

.88994 

.11  006 

.89  761 

11 

5O 

.78  772 

.89020 

.10980 

.89  752 

10 

51 

.78  788 

.89046 

.10954 

.89  742 

9 

52 

.78805 

.89073 

.10927 

.89  732 

8 

53 

.78821 

.89099 

.10901 

.89  722 

7 

54 

.78837 

.89  125 

.10875 

.89712 

6 

55 

.78853 

.89151 

.10849 

.89  702 

5 

56 

.78869 

.89177 

.10823 

.88693 

4 

57 

.78  886 

.89  203 

.10797 

.89683 

3 

58 

.78902 

.89  229 

.10771 

.89673 

2 

59 

.78918 

.89255 

.10745 

.89663 

1 

60 

.78934 

.89281 

.10719 

.89653 

0 

/ 

9  Lcos 

9Lcot 

1O  L  tan  9  L  sin 

/ 

/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

0 

.78934  .89281  .10719  .89653 

60 

1 

.78950  .89307  .10693  .89643 

59 

2 

.78967  .89333  .10667  .89633 

58 

3 

.78983  .89359  .10641  .89624 

57 

4 

.78999  .89385  .10615  .89614 

56 

5 

.79015  .89411  .10589  .89604 

55 

6 

.79031  .89437  .10563  .89594 

54 

7 

.79047  .89463  .10537  .89584 

53 

8 

.79063  .89489  .10511  .89574 

52 

9 

.79079  .89515  .10485  .89564 

51 

1O 

.79095  .89541  .10459  .89554 

5O 

11 

.79111  .89567  .10433  .89544 

49 

12 

.79128  .89593  .10407  .89534 

48 

13 

.79144  .89619  .10381  .89524 

47 

14 

.79160  .89645  .10355  .89514 

46 

15 

.79176  .89671  .10329  .89504 

45 

16 

.79192  .89697  .10303  .89495 

44 

17 

.79208  .89723  .10277  .89485 

43 

18 

.79224  .89749  .10251  .89475 

42 

19 

.79240  .89775  .10225  .89465 

41 

20 

.79256  .89801  .10199  .89455 

4O 

21 

.79272  .89827  .10173  .89445 

39 

22 

.79288  .89853  .10147  .89435 

38 

23 

.79304  .89879  .10121  .89425 

37 

24 

.79319  .89905  .10095  .89415 

36 

25 

.79335  .89931  .10069  .89405 

35 

26 

.79351  .89957  .10043  .89395 

34 

27 

.79367  .89983  .10017  .89385 

33 

28 

.79383  .90009  .09991  .89375 

32 

29 

.79399  .90035  .09965  .89364 

31 

3O 

.79415  .90061  .09939  .89354 

3O 

31 

.79431  .90086  .09914  .89344 

29 

32 

.79447  .90112  .09888  .89334 

28 

33 

.79463  .90138  .09862  .89324 

27 

34 

.79478  .90164  .09836  .89314 

26 

35 

.79494  .90190  .09810  .89304 

25 

36 

.79510  .90216  .09784  .89294 

24 

37 

.79526  .90242  .09758  .89284 

23 

38 

.79542  .90268  .09732  .89274 

22 

39 

.79558  .90294  .09706  .89264 

21 

4O 

.79573  .90320  .09680  .89254 

2O 

41 

.79589  .90346  .09654  .89244 

19 

42 

.79605  .90371  .09629  .89233 

18 

43 

.79621  .90397  .09603  .89223 

17 

44 

.79636  .90423  .09577  .89213 

16 

45 

.79652  .90449  .09551  .89203 

15 

46 

.79668  .90475  .09525  .89193 

14 

47 

.79684  .90501  .09499  .89183 

13 

48 

.79699  .90527  .09473  .89173 

12 

49 

.79715  .90553  .09447  .89162 

11 

50 

.79731  .90578  .09422  .89152 

1O 

51 

.79746  .90604  .09396  .89142 

9 

52 

.79762  .90630  .09370  .89132 

8 

53 

.79778  .90656  .09344  .89122 

7 

54 

.79793  .90682  .09318  .89112 

6 

55 

.79809  .90708  .09292  .89101 

5 

56 

.79825  .90734  .09266  .89091 

4 

57 

.79840  .90759  .09241  .89081 

3 

58 

.79856  .90785  .09215  .89071 

2 

59 

.79872  .90811  .09189  .89060 

1 

60 

.79887  .90837  .09163  .89050 

0 

/ 

9  Lcos  9Lcot  lOLtan  9Lsin 

/ 

52C 


51C 


39 


40' 


55 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

o 

.79887  .90837  .09163  .89050 

BO 

1 

.79903  .90863  .09137  .89040 

59 

2 

.79918  .90889  .09111  .89030 

58 

3 

.79934  .90914  .09086  .89020 

57 

4 

.79950  .90940  .09060  .89009 

56 

5 

.79965  .90966  .09034  .88999 

55 

6 

.79981  .90992  .09008  .88989 

54 

7 

.79996  .91018  .08982  .88978 

53 

8 

.80012  .91043  .08957  .88968 

52 

9 

.80027  .91069  .08931  .88958 

51 

10 

.80043  .91095  .08905  .88948 

50 

11 

.80058  .91121  .08879  .88937 

49 

12 

.80074  .91147  .08853  .88927 

48 

13 

.80089  .91172  .08828  .88917 

47 

14 

.80105  .91198  .08802  .88906 

46 

15 

.80120  .91224  .08776  .88896 

45 

16 

.80136  .91250  .08750  .88886 

44 

17 

.80151  .91276  .08724  .88875 

43 

18 

.80166  .91301  .08699  .88865 

42 

19 

.80182  .91327  .08673  .88855 

41 

2O 

.80197  .91353  .08647  .88844 

4O 

21 

.80213  .91379  .08621  .88834 

39 

22 

.80228  .91404  .08596  .88824 

38 

23 

.80244  .91430  .08570  .88813 

37 

24 

.80259  .91456  .08544  .88803 

36 

25 

.80274  .91482  .08518  .88793 

35 

26 

.80290  .91507  .08493  .88782 

34 

27 

.80305  .91533  .08467  .88772 

33 

28 

.80320  .91559  .08441  .88761 

32 

29 

.80336  .91585  .08415  .88751 

31 

30 

.80351  .91610  .08390  .88741 

3O 

31 

.80366  .91636  .08364  .88730 

29 

32 

.80382  .91662  .08338  .88720 

28 

33 

.80397  .91688  .08312  .88709 

27 

34 

.80412  .91713  .08287  .88699 

26 

35 

.80428  .91739  .08261  .88688 

25 

36 

.80443  .91765  .08235  .88678 

24 

37 

.80458  .91791  .08209  .88668 

23 

38 

.80473  .91816  .08184  .88657 

22 

39 

.80489  .91842  .08158  .88647 

21 

4O 

.80504  .91868  .08132  .88636 

20 

41 

.80519  .91893  .08107  .88626 

19 

42 

.80534  .91919  .08081  .88615 

18 

43 

.80550  .91945  .08055  .88605 

17 

44 

.80565  .91971  .08029  .88594 

16 

45 

.80580  .91996  .08004  .88584 

15 

46 

.80595  .92022  .07978  .88573 

14 

47 

.80610  .92048  .07952  .88563 

13 

48 

.80625  .92073  .07927  .88552 

12 

49 

.80641  .92099  .07901  .88542 

11 

50 

.80656  .92125  .07875  .88531 

10 

51 

.80671  .92150  .07850  .88521 

9 

52 

.80686  .92176  .07824  .88510 

8 

53 

.80701  .92202  .07798  .88499 

7 

54 

.80716  .92227  .07773  .88489 

6 

55 

.80731  .92253  .07747  .88478 

5 

56 

.80746  .92279  .07721  .88468 

4 

57 

.80762  .92304  .07696  .88457 

3 

58 

.80777  .92330  .07670  .88447 

2 

59 

.80792  .92356  .07644  .88436 

1 

60 

.80807  .92381  .07619  .88425 

O 

/ 

9Lcos  9Lcot  lOLtan  9Lsin 

/ 

/ 

9Lsin 

9Ltan 

10  L  cot 

9Lcos 

/ 

O 

.80807 

.92381 

.07  619 

.88425 

60 

1 

.80822 

.92407 

.07  593 

.88415 

59 

2 

.80837 

.92433 

.07  567 

.88404 

58 

3 

.80852 

.92458 

.07  542 

.88394 

57 

4 

.80867 

.92484 

.07516 

.88383 

56 

5 

.80882 

.92510 

.07  490 

.88372 

55 

6 

.80897 

.92  535 

.07  465 

.88362 

54 

7 

.80912 

.92561 

.07  439 

.88351 

53 

8 

.80927 

.92  587 

.07413 

.88340 

52 

9 

.80942 

.92612 

.07388 

.88330 

51 

10 

.80957 

.92638 

.07  362 

.88319 

50 

11 

.80972 

.92663 

.07  337 

.88308 

49 

12 

.80987 

.92  689 

.07311 

.88  298 

48 

13 

.81  002 

.92  715 

.07  285 

.88  287 

47 

14 

.81017 

.92  740 

.07  260 

.88276 

46 

15 

.81032 

.92  766 

.07  234 

.88  266 

45 

16 

.81  047 

.92  792 

.07  208 

.88255 

44 

17 

.81061 

.92817 

.07  183 

.88  244 

43 

18 

.81076 

.92  843 

.07157 

.88  234 

42 

19 

.81  091 

.92  868 

.07  132 

.88223 

41 

2O 

.81  106 

.92894 

.07  106 

.88212 

40 

21 

.81  121 

.92920 

.07080 

.88  201 

39 

22 

.81  136 

.92  945 

.07055 

.88  191 

38 

23 

.81151 

.92971 

.07029 

.88  180 

37 

24 

.81  166 

.92996 

.07004 

.88  169 

36 

25 

.81  180 

.93022 

.06978 

.88  158 

35 

26 

.81  195 

.93048 

.06952 

.88  148 

34 

27 

.81210 

.93073 

.06927 

.88137 

33 

28 

.81  225 

.93099 

.06901 

.88126 

32 

29 

.81  240 

.93  124 

.06876 

.88115 

31 

3O 

.81  254 

.93  150 

.06850 

.88  105 

3O 

31 

.81269 

.93175 

.06825 

.88094 

29 

32 

.81  284 

.93  201 

.06  799 

.88083 

28 

33 

.81  299 

.93  227 

.06  773 

.88072 

27 

34 

.81314 

.93252 

.06748 

.88061 

26 

35 

.81328 

.93  278 

.06  722 

.88051 

25 

36 

.81  343 

.93  303 

.06697 

.88040 

24 

37 

.81358 

.93329 

.06671 

.88029 

23 

38 

.81372 

.93  354 

.06646 

.88018 

22 

39 

.81387 

.93380 

.06620 

.88007 

21 

4O 

.81  402 

.93  406 

.06594 

.87996 

20 

41 

.81417 

.93431 

.06569 

.87985 

19 

42 

.81431 

.93457 

.06543 

.87975 

18 

43 

.81  446 

.93  482 

.06518 

.87964 

17 

44 

.81461 

.93  508 

.06492 

.87953 

16 

45 

.81  475 

.93  533 

.06467 

.87942 

15 

46 

.81  490 

.93  559 

.06441 

.87931 

14 

47 

.81  505 

.93  584 

.06416 

.87  920 

13 

48 

.81519 

.93610 

.06390 

.87  909 

12 

49 

.81  534 

.93  636 

.06364 

.87  898 

11 

50 

.81  549 

.93  661 

.06339 

.87  887 

1O 

51 

.81  563 

.93  687 

.06313 

.87877 

9 

52 

.81578 

.93  712 

.06288 

.87  866 

8 

53 

.81  592 

.93  738 

.06262 

.87855 

7 

54 

.81  607 

.93  763 

.06237 

.87844 

6 

55 

.81622 

.93  789 

.06211 

.87  833 

5 

56 

.81636 

.93  814 

.06  186 

.87822 

4 

57 

.81651 

.93  840 

.06  160 

.87811 

3 

58 

.81  665 

.93  865 

.06  135 

.87  800 

2 

59 

.81680 

.93  891 

.06  109 

.87  789 

1 

60 

.81  694 

.93  916 

.06084 

.87  778 

0 

r 

9Lcos 

9Lcot 

lOLtan  9Lsin 

' 

50C 


49' 


41C 


42C 


/ 

9Lsin  9Ltan  lOLcot  9Lcos      / 

o 

.81694  .93916  .06084  .87778 

GO 

1 

.81709  .93942  .06058  .87767 

59 

2 

.81723  .93967  .06033  .87756 

58 

3 

.81738  .93993  .06007  .87745 

57 

4 

.81752  .94018  .05982  .87734 

56 

5 

.81767  .94044  .05956  .87723 

55 

6 

.81781   .94069  .05931   .87712 

54 

*7 

.81796  .94095   .05905   .87701 

53 

8 

.81810  .94120  .05880  .87690 

52 

9 

.81825   .94146  .05854  .87679 

51 

10 

.81839  .94171  .05829  .87668 

50 

11 

.81854  .94197  .05803  .87657 

49 

12 

.81868  .94222  .05778  .87646 

48 

13 

.81882  .94248  .05752  .87635 

47 

14 

.81897  .94273   .05727  .87624 

46 

15 

.81911  .94299  .05701  .87613 

45 

16 

.81926  .94324  .05676  .87601 

44 

17 

.81940  .94350  .05650  .87590 

43 

18 

.81955   .94375  .05625  .87579 

42 

19 

.81969  .94401  .05599  .87568 

41 

20 

.81983  .94426  .05574  .87557 

40 

21 

.81998  .94452  .05548  .87546 

39 

22 

.82012  .94477  .05523  .87535 

38 

23 

.82026  .94503  .05497  .87524 

37 

24 

.82041  .94528  .05472  .87513 

36 

25 

.82055   .94554  .05446  .87501 

35 

26 

.82069  .94579  .05421   .87490 

34 

27 

.82084  .94604  .05396  .87479 

33 

28 

.82098  .94630  .05370  .87468 

32 

29 

.82112  .94655  .05345   .87457 

31 

3O 

.82126  .94681   .05319  .87446 

3O 

31 

.82141  .94706  .05294  .87434 

29 

32 

.82155   .94732  .05268  .87423 

28 

33 

.82169  .94757  .05243  .87412 

27 

34 

.82184  .94783  .05217  .87401 

26 

35 

.82198  .94808  .05192  .87390 

25 

36 

.82212  .94834  .05166  .87378 

24 

37 

.82226  .94859  .05141   .87367 

23 

38 

.82240  .94884  .05116  .87356 

22 

39 

.82255   .94910  .05090  .87345 

21 

4O 

.82269  .94935   .05065   .87334 

20 

41 

.82283  .94961   .05039  .87322 

19 

42 

.82297  .94986  .05014  .87311 

18 

43 

.82311   .95012  .04988  .87300 

17 

44 

.82326  .95037  .04963  .87288 

16 

45 

.82340  .95062  .04938  .87277 

15 

46 

.82354  .95088  .04912  .87266 

14 

47 

.82368  .95113   .04887  .87255 

13 

48 

.82382  .95139  .04861  .87243 

12 

49 

.82396  .95164  .04836  .87232 

11 

5O 

.82410  .95190  .04810  .87221 

1O 

51 

.82424  .95215  .04785   .87209 

9 

52 

.82439  .95240  .04760  .87198 

8 

53 

.82453  .95266  .04734  .87187 

7 

54 

.82467  .95291   .04709  .87175 

6 

55 

.82481   .95317  .04683  .87164 

5 

56 

.82495   .95342  .04658  .87153 

4 

57 

.82509  .95368  .04632  .87141 

3 

58 

.82523   .95393  .04607  .87130 

2 

59 

.82537  .95418  .04582  .87119 

1 

60 

.82551   .95444  .04556  .87107 

0 

/ 

9Lcos  9LcotlOLtan9Lsin 

/ 

/ 

9Lsin 

9Ltan 

lOLcot  9  Lcos  |_/_ 

O 

.82551 

.95  444 

.04  556 

.87  107 

GO 

1 

.82  565 

.95  469 

.04531 

.87  096 

59 

2 

.82579 

.95  495 

.04  505 

.87085 

58 

3 

.82  593 

.95  520 

.04480 

.87073 

57 

4 

.82607 

.95  545 

.04  455 

.87062 

56 

5 

.82621 

.95  571 

.04  429 

.87050 

55 

6 

.82635 

.95  596 

.04  404 

.87  039 

54 

7 

.82649 

.95  622 

.04378 

.87028 

53 

8 

.82  663 

.95647 

.04353 

.87016 

52 

9 

.82677 

.95  672 

.04328 

.87  005 

51 

1O 

.82691 

.95  698 

.04  302 

.86993 

50 

11 

.82  705 

.95  723 

.04  277 

.86982  i  49 

12 

.82719 

.95  748 

.04  252 

.86970  48 

13 

.82  733 

.95  774 

.04  226 

.86959  47 

14 

.82  747 

.95  799 

.04  201 

.86947 

46 

15 

.82  761 

.95  825 

.04  175 

.86936 

45 

16 

.82775 

.95  850 

.04  150 

.86924 

44 

17 

.82  788 

.95  875 

.04  125 

.86913 

43 

18 

.82  802 

.95  901 

.04  099 

.86902 

42 

19 

.82816 

.95  926 

.04074 

.86890 

41 

20 

.82830 

.95952 

.04048 

.86879 

4O 

21 

.82  844 

.95  977 

.04023 

.86867 

39 

22 

.82858 

.96002 

.03  998 

.86855 

38 

23 

.82872 

.96028 

.03  972 

.86844 

37 

24 

.82885 

.96  053 

.03947 

.86832 

36 

25 

.82  899 

.96078 

.03  922 

.86821 

35 

26 

.82913 

.96  104 

.03  896 

.86809 

34 

27 

.82927 

.96129 

.03871 

.86  798 

33 

28 

.82941 

.96  155 

.03  845 

.86  786 

32 

29 

.82955 

.96  180 

.03  820 

.86  775 

31 

30 

.82968 

.96205 

.03  795 

.86  763 

30 

31 

.82982 

.96231 

.03  769 

.86752 

29 

32 

.82996 

.96256 

.03  744 

.86  740 

28 

33 

.83  010 

.96  281 

.03  719 

.86  728 

27 

34 

.83  023 

.96307 

.03  693 

.86717 

26 

35 

.83  037 

.96332 

.03  668 

.86705 

25 

36 

.83051 

.96357 

.03  643 

.86694 

24 

37 

.83  065 

.96383 

.03617 

.86682 

23 

38 

.83  078 

.96408 

.03  592 

.86670 

22 

39 

.83  092 

.96433 

.03  567 

.86659 

21 

40 

.83  106 

.96459 

.03  541 

.86647 

20 

41 

.83120 

.96484 

.03516 

.86635 

19 

42 

.83  133 

.96510 

.03  490 

.86624 

18 

43 

.83  147 

.96535 

.03465 

.86612 

17 

44 

.83  161 

.96560 

.03  440 

.86600 

16 

45 

.83  174 

.96  586 

.03  414 

.86589 

15 

46 

.83  188 

.96611 

.03389 

.86577 

14 

47 

.83  202 

.96636 

.03  364 

.86  565 

13 

48 

.83215 

.96662 

.03  338 

.86554 

12 

49 

.83  229 

.96687 

.03313 

.86542 

11 

50 

.83  242 

.96712 

.03  288 

.86530 

1O 

51 

.83  256 

.96  738 

.03  262 

.86518 

9 

52 

.83  270 

.96  763 

.03  237 

.86  507 

8 

53 

.83  283 

.96  788 

.03  212 

.86495 

7 

54 

.83  297 

.96814 

.03  186 

.86483 

6 

55 

.83310 

.96839 

.03  161 

.86472 

5 

56 

.83  324 

.96864 

.03  136 

.86460 

4 

57 

.83338 

.96890 

.03  110 

.86448 

3 

58 

.83351 

.96915 

.03085 

.86436 

2 

59 

.83365 

.96940 

.03  060 

.86425 

1 

6O 

.83  378 

.96966 

.03034 

.86413 

0 

/ 

9Lcos 

9LcotlOLtan 

9  L  sin   / 

48' 


47° 


43' 


44C 


57 


/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

o 

.83378  .96966  .03034  .86413 

W 

1 

.83392  .96991  .03009  .86401 

59 

2 

.83405  .97016  .02984  .86389 

58 

3 

.83419  .97042  .02958  .86377 

57 

4 

.83.432  .97067  .02933  .86366 

56 

5 

.83446  .97092  .02908  .86354 

55 

6 

.83459  .97118  .02882  .86342 

54 

7 

.83473  .97143  .02857  .86330 

53 

8 

.83486  .97168  .02832  .86318 

52 

9 

.83500  .97193  .02807  .86306 

51 

1C 

.83513  .97219  .02781  .86295 

50 

11 

.83527  .97244  .02756  .86283 

49 

12 

.83540  .97269  .02731  .86771 

48 

13 

.83554  .97295  .02705  .86259 

47 

14 

.83567  .97320  .02680  .86247 

46 

IS 

.83581  .97345  .02655  .86235 

45 

16 

.83594  .97371  .02629  .86223 

44 

17 

.83608  .97396  .02604  .86211 

43 

18 

.83621  .97421  .02579  .86200 

42 

19 

.83634  .97447  .02553  .86188 

41 

2O 

.83648  .97472  .02528  .86176 

40 

21 

.83661  .97497  .02503  .86164 

39 

22 

.83674  .97523  .02477  .86152 

38 

23 

.83688  .97548  .02452  .86140 

37 

24 

.83701  .97573  .02427  .86128 

36 

25 

.83715  .97598  .02402  .86116 

35 

26 

.83728  .97624  .02376  .86104 

34 

27 

.83741  .97649  .02351  .86092 

33 

28 

.83755  .97674  .02326  .86080 

32 

29 

.83768  .97700  .02300  .86068 

31 

30 

.83781  .97725  .02275  .86056 

30 

31 

.83795  .97750  .02250  .86044 

29 

32 

.83808  .97776  .02224  .86032 

28 

33 

.83821  .97801  .02199  .86020 

27 

34 

.83834  .97826  .02174  .86008 

26 

35 

.83848  .97851  .02149  .85996 

25 

36 

.83861  .97877  .02123  .85984 

24 

37 

.83874  .97902  .02098  .85972 

23 

38 

.83887  .97927  .02073  .85960 

22 

39 

.83901  .97953  .02047  .85948 

21 

4O 

.83914  .97978  .02022  .85936 

2O 

41 

.83927  .98003  .01997  .85924 

19 

42 

.83940  .98029  .01971  .85912 

18 

43 

.88954  .98054  .01946  .85900 

17 

44 

.83967  .98079  .01921  .85888 

16 

45 

.83980  .98104  .01896  .85876 

15 

46 

.83993  .98130  .01870  .85864 

14 

47 

.84006  .98155  .01845  .85851 

13 

48 

.84020  .98180  .01820  .85839 

12 

49 

.84033  .98206  .01794  .85827 

11 

50 

.84046  .98231  .01769  .85815 

1O 

51 

.84059  .98256  .01744  .85803 

9 

52 

.84072  .98281  .01719  .85791 

8 

53 

.84085  .98307  .01693  .85779 

7 

54 

.84098  .98332  .01668  .85766 

6 

55 

.84112  .98357  .01643  .85754 

5 

56 

.84125  .98383  .01617  .85742 

4 

57 

.84138  .98408  .01592  .85730 

3 

58 

.84151  .98433  .01567  .85718 

2 

59 

.84164  .98458  .01542  .85706 

1 

60 

.84177  .98484  .01516  .85693 

0 

/ 

9Lcos  9Lcot  10  L  tan  9Lsin 

/ 

/ 

9  L  sin  9  L  tan  1O  L  cot  9  L  cos 

/ 

~cT 

.84177  .98484  .01516  .85693 

6O 

i 

.84190  .98509  .01491  .85681 

59 

2 

.84203   .98534   .01466  .85669 

58 

3 

.84216  .98560  .01440  .85657 

57 

4 

.84229  .98585   .01415   .85645 

56 

5 

.84242  .98610  .01390  .85632 

55 

6 

.84255   .98635   .01365   .85620 

54 

.7 

.84269  .98661  .01339  .85608 

53 

8 

.84282   .98686  .01314  .85596 

52 

9 

.84295    .98711  .01289  .85583 

51 

1O 

.84308  .98737  .01263  .85571 

50 

11 

.84321   .98762  .01238  .85559 

49 

12 

.84334  .98787  .01213  .85547 

48 

13 

.84347  .98812  .01188  .85534 

47 

14 

.84360  .98838  .01162  .85522 

46 

15 

.84373   .98863  .01137  .85510 

45 

16 

.84385   .98888  .01112  .85497 

44 

17 

.84398  .98913   .01087  .85485 

43 

18 

.84411   .98939  .01061  .85473 

42 

19 

.84424  .98964  .01036  .85460 

41 

2O 

.84437  .98989  .01011   .85448 

4O 

21 

.84450  .99015   .00985   .85436 

39 

22 

.84463   .99040  .00960  .85423 

38 

23 

.84476  .99065   .00935   .85411 

37 

24 

.84489  .99090  .00910  .85399 

36 

25 

.84502  .99116  .00884  .85386 

35 

26 

.84515   .99141   .00859  .85374 

34 

27 

.84528  .99166  .00834  .85361 

33 

28 

.84540  .99191   .00809  .85349 

32 

29 

.84553  .99217  .00783  .85337 

31 

30 

.84566  .99242  .00758  .85324 

3O 

31 

.84579  .99267  .00733  .85312 

29 

32 

.84592  .99293   .00707  .85299 

28 

33 

.84605   .99318  .00682  .85287 

27 

34 

.84618  .99343   .00657  .85274 

26 

35 

.84630  .99368  .00632  .85262 

25 

36 

.84643   .99394  .00606  .85250 

24 

37 

.84656  .99419  .00581   .85237 

23 

38 

.84669  .99444  .00556  .85225 

22 

39 

.84682  .99469  .00531  .85212 

21 

40 

.84694  .99495   .00505   .85200 

20 

41 

.84707  .99520  .00480  .85187 

19 

42 

.84720  .99545   .00455   .85175 

18 

43 

.84733   .99570  .00430  .85162 

17 

44 

.84745   .99596  .00404  .85150 

16 

45 

.84758  .99621  .00379  .85137 

15 

46 

.84771   .99646  .00354  .85125 

14 

47 

.84784  .99672  .00328  .85112 

13 

48 

.84796  .99697   .00303  .85100 

12 

49 

.84809  .99722  .00278  .85087 

11 

50 

.84822  .99747  .00253  .85074 

1O 

51 

.84835   .99773   .00227  .85062 

9 

52 

.84847   .99798  .00202  .85049 

8 

53 

.84860  .99823   .00177  .85037 

7 

54 

.84873  .99848  .00152  .85024 

6 

55 

.84885   .99874   .00126  .85012 

5 

56 

.84898  .99899  .00101   .84999 

4 

57 

.84911   .99924   .00076  .84986 

3 

58 

.84923   .99949   .00051   .84974 

2 

59 

.84936  .99975   .00025   .84961 

1 

6O 

.84949  .00000  .00000  .S49J2_. 

O 

/ 

9  L  cos  1O  L  cot  1O  L  tan  9~lTsiii 

/ 

46' 


45< 


58 


TABLE  IV  — NATURAL  FUNCTIONS 


0° 

/ 

sin        tan        cot        cos 

/ 

o 

.00000  .00000      co        1.0000 

60 

1 

.00029  .00029  3437.7    1.0000 

59 

2 

.00058  .00058  1718.9    1.0000 

58 

3 

.00087  .00087  1145.9    1.0000 

57 

4 

.00116  .00116  859.44    1.0000 

56 

5 

.00  145  .00  145  687.55    1.0000 

55 

6 

.00  175  .00175  572.96    1.0000 

54 

7 

.00204  .00204  491.11    1.0000 

53 

8 

.00233  .00233  429.72    1.0000 

52 

9 

.00262  .00262  381.97    1.0000 

51 

1C 

.00291  .00291  343.77    1.0000 

5O 

11 

.00320  .00320  312.52  .99999 

49 

12 

.00349  .00349  286.48  .99999 

48 

13 

.00378  .00378  264.44  .99999 

47 

14 

.00407  .00407  245.55  .99999 

46 

15 

.00436  .00436  229.18  .99999 

45 

16 

.00465  .00465  214.86  .99999 

44 

17 

.00495  .00495  202.22  .99999 

43 

18 

.00524  .00524  190.98  .99999 

42 

19 

.00553  .00553  180.93  .99998 

41 

20 

.00  582  -.00  582  171.89  .99  998 

4O 

21 

.00611  .00611  163.70  .99998 

39 

22 

.00640  .00640  156.26  .99998 

38 

23 

.00669  .00669  149.47  .99998 

37 

24 

.00698  .00698  143.24  .99998 

36 

25 

.00727  .00727  137.51  .99997 

35 

26 

.00756  .00756  132.22  .99997 

34 

27 

.00785  .00785  127.32  .99997 

33 

28 

.00814  .00815  122.77  .99997 

32 

29 

.00844  .00844  118.54  .99996 

31 

30 

.00873  .00873  114.59  .99996 

3O 

31 

.00902  .00902  110.89  .99996 

29 

32 

.00931  .00931  107.43  .99996 

28 

33 

.00960  .00960  104.17  .99995 

27 

34 

.00989  .00989  101.11  .99995 

26 

35 

.01018  .01018  98.218  .99995 

25 

36 

.01047  .01047  95.489  .99995 

24 

37 

.01  076  .01  076  92.908  .99  994 

23 

38 

.01105  .01  105  90.463  .99994 

22 

39 

.01  134  .01  135  88.144  .99994 

21 

40 

.01  164  .01  164  85.940  .99993 

20 

41 

.01  193  .01  193  83.844  .99  993 

19 

42 

.01222  .01222  81.847  .99993 

18 

43 

.01251  .01251  79.943  .99992 

17 

44 

.01280  .01280  78.126  .99992 

16 

45 

.01  309  .01  309  76.390  .99  991 

15 

46 

.01338  .01338  74.729  .99991 

14 

47 

.01367  .01367  73.139  .99991 

13 

48 

.01396  .01396  71.615  .99990 

12 

49 

.01425  .01425  70.153  .99990 

11 

50 

.01  454  .01  455  68.750  .99  989 

10 

51 

.01483  .01484  67.402  .99989 

9 

52 

.01513  .01  513  66.105  .99989 

8 

53 

.01542  .01  542  64.858  .99988 

7 

54 

.01571  .01571  63.657  .99988 

6 

55 

.01  600  .01  600  62.499  .99987 

5 

56 

.01629  .01629  61.383  .99987 

4 

57 

.01  658  .01  658  60.306  .99  986 

3 

58 

.01  687  .01  687  59.266  .99  986 

2 

59 

.01716  .01  716  58.261  .99985 

1 

6O 

.01  745  .01  746  57.290  .99985 

O 

/ 

cos        cot        tan        sin 

/ 

89° 

1° 

/ 

sin        tan        cot        cos 

/ 

O 

.01  745  .01  746  57.290  .99  985 

00 

1 

.01774  .01775  56.351  .99984 

59 

2 

.01803  .01804  55.442  .99984 

58 

3 

.01832  .01833  54.561  .99983 

57 

4 

.01862  .01862  53.709  .99983 

56 

5 

.01891  .01891  52.882  .99982 

55 

6 

.01920  .01  920  52.081  .99982 

54 

7 

.01949  .01949  51.303  .99981 

53 

8 

.01978  .01978  50.549  .99980 

52 

9 

.02007  .02007  49.816  .99980 

51 

1O 

.02036  .02036  49.104  .99979 

50 

11 

.02065  .02066  48.412  .99979 

49 

12 

.02094  .02095  47.740  .99978 

48 

13 

.02123  .02124  47.085  .99977 

47 

14 

.02152  .02153  46.449  .99977 

46 

15 

.02181  .02182  45.829  .99976 

45 

16 

.02211  .02211  45.226  .99976 

44 

17 

.02  240  .02  240  44.639  .99  975 

43 

18 

.02  269  .02  269  44.066  .99  974 

42 

19 

.02298  .02298  43.508  .99974 

41 

2O 

.02327  .02328  42.964  .99973 

40 

21 

.02356  .02357  42.433  .99972 

39 

22 

.02385  .02386  41.916  .99972 

38 

23 

.02414  .02415  41.411  .99971 

37 

24 

.02443  .02444  40.917  .99970 

36 

25 

.02  472  .02  473  40.436  .99  969 

35 

26 

.02501  .02502  39.965  .99969 

34 

27 

.02530  .02531  39.506  .99968 

33 

28 

.02560  .02560  39.057  .99967 

32 

29 

.02589  .02589  38.618  .99966 

31 

3O 

.02618  .02619  38.188  .99966 

30 

31 

.02647  .02648  37.769  .99965 

29 

32 

.02676  .02677  37.358  .99964 

28 

33 

.02  705  .02  706  36.956  .99  963 

27 

34 

.02  734  .02  735  36.563  .99  963 

26 

35 

.02763  .02764  36.178  .99962 

25 

36 

.02792  .02793  35.801  .99961 

24 

37 

.02821  .02822  35.431  .99960 

23 

38 

.02850  .02851  35.070  .99959 

22 

39 

.02879  .02881  34.715  .99959 

21 

40 

.02908  .02910  34.368  .99958 

20 

41 

.02938  .02939  34.027  .99957 

19 

42 

.02967  .02968  33.694  .99956 

18 

43 

.02996  .02997  33.366  .99955 

17 

44 

.03  025  .03  026  33.045  .99  954 

16 

45 

.03054  .03055  32.730  .99953 

15 

46 

.03083  .03084  32.421  .99952 

14 

47 

.03112  .03114  32.118  .99952 

13 

48 

.03141  .03143  31.821  .99951 

12 

49 

.03  170  .03  172  31.528  .99950 

11 

50 

.03199  .03201  31.242  .99949 

10 

51 

.03  228  .03  230  30.960  .99  948 

9 

52 

.03257  .03259  30.683  .99947 

8 

53 

.03  286  .03  288  30.412  .99  946 

7 

54 

.03316  .03317  30.145  .99945 

6 

55 

.03345  .03346  29.882  .99944 

5 

56 

.03374  .03376  29.624  .99943 

4 

57 

.03403  .03405  29.371  .99942 

3 

58 

.03432  .03434  29.122  .99941 

2 

59 

.03461  .03463  28.877  .99940 

1 

GO 

.03  490  .03  492  28.636  .99  939 

O 

/ 

cos        cot        tan        sin 

/ 

88° 

NATURAL   FUNCTIONS 


59 


2° 

/ 

sin   tan   cot   cos 

/ 

0 

.03490  .03492  28.636  .99939 

60 

1 

.03519  .03521  28.399  .99938 

59 

2 

.03548  .03550  28.166  .99937 

58 

3 

.03577  .03579  27.937  .99936 

57 

4 

.03  606  .03  609  27.712  .99  935 

56 

5 

.03  635  .03  638  27.490  .99  934 

55 

6 

.03664  .03667  27.271  .99933 

54 

7 

.03693  .03696  27.057  .99932 

53 

8 

.03723  .03725  26.845  .99931 

52 

9 

.03  752  .03  754  26.637  .99  930 

51 

1C 

.03  781  .03  783  26.432  .99  929 

5O 

11 

.03810  .03812  26.230  .99927 

49 

12 

.03839  .03842  26.031  .99926 

48 

13 

.03868  .03871  25.835  .99925 

47 

14 

.03897  .03900  25.642  .99924 

46 

15 

.03  926  .03  929  25.452  .99  923 

45 

16 

.03955  .03958  25.264  .99922 

44 

17 

.03984  .03987  25.080  .99921 

43 

18 

.04013  .04016  24.898  .99919 

42 

19 

.04042  .04046  24.719  .99918 

41 

2O 

.04071  .04075  24.542  .99917 

4O 

21 

.04  100  .04  104  24.368  .99  916 

39 

22 

.04129  .04133  24.196  .99915 

38 

23 

.04  159  .04  162  24.026  .99913 

37 

24 

.04188  .04191  23.859  .99912 

36 

25 

.04217  .04220  23.695  .99911 

35 

26 

.04  246  .04  250  23.532  .99  910 

34 

27 

.04  275  .04  279  23.372  .99  909 

33 

28 

.04304  .04308  23.214  .99907 

32 

29 

.04333  .04337  23.058  .99906 

31 

3O 

.04362  .04366  22.904  .99905 

30 

31 

.04391  .04395  22.752  .99904 

29 

32 

.04420  .04424  22.602  .99902 

28 

33 

.04449  .04454  22.454  .99901 

27 

34 

.04478  .04483  22.308  .99900 

26 

35 

.04507  .04512  22.164  .99898 

25 

36 

.04536  .04541  22.022  .99897 

24 

37 

.04565  .04570  21.881  .99896 

23 

38 

.04  594  .04  599  21.743  .99  894 

22 

39 

.04623  .04628  21.606  .99893 

21 

40 

.04653  .04658  21.470  .99892 

2O 

41 

.04682  .04687  21.337  .99890 

19 

42 

.04711  .04716  21.205  .99889 

18 

43 

.04  740  .04  745  21.075  .99  888 

17 

44 

.04  769  .04  774  20.946  .99  886 

16 

45 

.04  798  .04  803  20.819  .99  885 

15 

46 

.04827  .04833  20.693  .99883 

14 

47 

.04856  .04862  20.569  .99882 

13 

48 

.04885  .04891  20.446  .99881 

12 

49 

.04914  .04920  20.325  .99879 

11 

5O 

.04943  .04949  20.206  .99878 

1O 

51 

.04  972  .04  978  20.087  .99  876 

9 

52 

.05  001  .05  007  19.970  .99  875 

8 

53 

.05  030  .05  037  19.855  .99  873 

7 

54 

.05  059  .05  066  19.740  .99  872 

6 

55  i  .05  OSS  .05  095  19.627  .99  870 

5 

56 

.05117  .05124  19.516  .99869 

4 

57 

.05  146  .05  153  19.405  .99867 

3 

58 

.05  175  .05  182  19.296  .99866 

2 

59 

.05  205  .05  212  19.188  .99  864 

1 

60 

.05  234  .05  241  19.081  .99  863 

0 

/ 

cos    cot   tan    sin 

/ 

87° 

3° 

/ 

sin   tan   cot   cos 

/ 

0 

.05  234  .05  241  19.081  .99  863 

6O 

1 

.05  263  .05  270  18.976  .99  861 

59 

2 

.05  292  .05  299  18.871  .99  860 

58 

3 

.05  321  .05  328  18.768  .99  858 

57 

4 

.05350  .05357  18.666  .99857 

56 

5 

.05  379  .05  387  18.564  .99  855 

55 

6 

.05408  .05416  18.464  .99854 

54 

7 

.05437  .05445  18.366  .99852 

53 

8 

.05466  .05474  18.268  .99851 

52 

9 

.05495  .05503  18.171  .99849 

51 

1O 

.05  524  .05  533  18.075  .99  847 

50 

11 

.05  553  .05  562  17.980  .99  846 

49 

12 

.05582  .05591  17.886  .99844 

48 

13 

.05611  .05620  17.793  .99842 

47 

14 

.05  640  .05  649  17.702  .99  841 

46 

15 

.05669  .05678  17.611  .99839 

45 

16 

.05.698  ,05  708  17.521  £a83& 

44 

17 

.05727  .05737  17.431  .99836 

43 

18 

.05  756  .05  766  17.343  .99834 

42 

19 

.05785  .05795  17.256  .99833 

41 

20 

.05  814  .05  824  17.169  .99  831 

4O 

21 

.05  844  .05  854  17.084  .99  829 

39 

22 

.05  873  .05  883  16.999  .99  827 

38 

23 

.05902  .05912  16.915  .99826 

37 

24 

.05931  .05941  16.832  .99824 

36 

25 

.05  960  .05  970  16.750  .99  822 

35 

26 

.05989  .05999  16.668  .99821 

34 

27 

.06018  .06029  16.587  .99819 

33 

28 

.06047  .06058  16.507  .99817 

32 

29 

.06076  .06087  16.428  .99815 

31 

30 

.06105  .06116  16.350  .99813 

3O 

31 

.06  134  .06  145  16.272  .99  812 

29 

32 

.06163  .06175  16.195  .99810 

28 

33 

.06192  .06204  16.119  .99808 

27 

34 

.06221  .06233  16.043  .99806 

26 

35 

.06250  .06262  15.969  .99804 

25 

36 

.06279  .06291  15.895  .99803 

24 

37 

.06308  .06321  15.821  .99801 

23 

38 

.06337  .06350  15.748  .99799 

22 

39 

.06  366  .06  379  15.676  .99  797 

21 

4O 

.06395  .06408  15.605  .99795 

20 

41 

.06424  .06438  15.534  .99793 

19 

42 

.06453  .06467  15.464  .99792 

18 

43 

.06482  .06496  15.394  .99790 

17 

44 

.06511  .06525  15.325  .99788 

16 

45 

.06  540  .06  554  15.257  .99  786 

15 

46 

.06569  .06584  15.189  .99784 

14 

47 

.06  59S  .06  613  15.122  .99  782 

13 

48 

.06627  .06642  15.056  .99780 

12 

49 

.06  656  .06  671  14.990  .99  778 

11 

50 

.06685  .06700  14.924  .99776 

10 

51 

.06714  .06730  14.860  .99774 

9 

52 

.06743  .06759  14.795  .99772 

8 

53 

.06  773  .06  788  14.732  .99  770 

7 

54 

.06802  .06817  14.669  .99768 

6 

55 

.06831  .06847  14.606  .99766 

5 

56 

.06  860  .06  876  14.544  .99  764 

4 

57 

.06889  .06905  14.482  .99762 

3 

58 

.06918  .06934  14.421  .99760 

2 

59 

.06947  .06963  14.361  .99758 

1 

6O 

.06  976  .06  993  14.301  .99  756 

O 

/ 

cos    cot   tan    sin 

/ 

86° 

60 


NATURAL   FUNCTIONS 


4° 

/ 

sin    tan    cot    cos 

/ 

o 

.06976  .06993  14.301  .99756 

6O 

1 

.07005  .07022  14.241  .99754 

59 

2 

.07034  .07051  14.182  .99752 

58 

3 

.07063  .07080  14.124  .99750 

57 

4 

.07  092  .07  110  14.065  .99  748 

56 

5 

.07  121  .07  139  14.008  .99  746 

55 

6 

.07  150  .07  168  13.951  .99  744 

54 

7 

.07  179  .07  197  13.894  .99  742 

53 

8 

.07208  .07227  13.838  .99740 

52 

9 

.07  237  .07  256  13.782  .99  738 

51 

10 

.07  266  .07  285  13.727  .99  736 

50 

11 

.07295  .07314  13.672  .99734 

49 

12 

.07324  .07344  13.617  .99731 

48 

13 

.07353  .07373  13.563  .99729 

47 

14 

.07382  .07402  13.510  .99727 

46 

15 

.07411  .07431  13.457  .99725 

45 

16 

.07  440  .07  461  13.404  .99  723 

44 

17 

.07  469  .07  490  13.352  .99  721 

43 

18 

.07498  .07519  13.300  .99719 

42 

19 

.07527  .07548  13.248  .99716 

41 

20 

.07556  .07578  13.197  .99714 

4O 

21 

.07585  .07607  13.146  .99712 

39 

22 

.07  614  .07  636  13.096  .99  710 

38 

23 

.07  643  .07  665  13.046  .99  708 

37 

24 

.07  672  .07  695  12.996  .99  705 

36 

25 

.07  701  .07  724  12.947  .99  703 

35 

26 

.07  730  .07  753  12.898  .99  701 

34 

27 

.07  759  .07  782  12.850  .99  699 

33 

28 

.07  788  .07  812  12  801  .99  696 

32 

29 

.07817  .07841  12.754  .99694 

31 

30 

.07846  .07870  12.706  .99692 

3O 

31 

.07  875  .07  899  12.659  .99  689 

29 

32 

.07904  .07929  12.612  .99687 

28 

33 

.07933  .07958  12.566  .99685 

27 

34 

.07962  .07987  12.520  .99683 

26 

35 

.07991  .08017  12.474  .99680 

25 

36 

.08020  .08046  12.429  .99678 

24 

37 

.080*9  .08075  12.384  .99676 

23 

38 

.08  078  .08  104  12.339  .99  673 

22 

39 

.08107  .08134  12.295  .99671 

21 

40 

.08136  .08163  12.251  .99668 

20 

41 

.08  165  .08  192  12.207  .99  666 

19 

42 

.08  194  .08  221  12.163  .99  664 

18 

43 

.08223  .08251  12.120  .99661 

17 

44 

.08252  .08280  12.077  .99659 

16 

45 

.08  281  .08  309  12.035  .99  657 

15 

46 

.08310  .08339  11.992  .99654 

14 

47 

.08339  .08368  11.950  .99652 

13 

48 

.08368  .08397  11.909  .99649 

12 

49 

.08397  .08427  11.867  .99647 

11 

50 

.08426  .08456  11.826  .99644 

1O 

51 

.08455  .08485  11.785  .99642 

9 

52 

.08484  .08514  11.745  .99639 

8 

53 

.08513  .08544  11.705  .99637 

7 

54 

.08542  .08573  11.664  .99635 

6 

55 

.08571  .08602  11.625  .99632 

5 

56 

.08600  .08632  11.585  .99630 

4 

57 

.08629  .08661  11.546  .99627 

3 

58 

.08658  .08690  11.507  .99625 

2 

59 

.08687  .08720  11.468  .99622 

1 

60 

.08716  .08749  11.430  .99619 

0 

/ 

cos    cot   tan    sin 

/ 

85° 

5° 

/ 

sin        tan        cot        cos 

/ 

O 

.08  716  .08  749  11.430  .99  619 

TK> 

1 

.08745  .08778  11.392  .99'  617 

59 

2 

.08774  .08807  11.354  .99614 

58 

3 

.08803  .08837  11.316  .99612 

57 

4 

.08831  .08866  11.279  .99609 

56 

5 

.08860  .08895  11.242  .99607 

55 

6 

.08889  .08925  11.205  .99604 

54 

7 

.08918  .08954  11.168  .99602 

.53 

8 

.08947  .08983  11.132  .99599 

52 

9 

.08976  .09013  11.095  .99596 

51 

10 

.09005  .09042  11.059  .99594 

5O 

11 

.09034  .09071  11.024  .99591 

49 

12 

.09  063  .09  101  10.988  .99  588 

48 

13 

.09092  .09130  10.953  .99586 

47 

14 

.09  121  .09  159  10.918  .99  583 

46 

15 

.09150  .09189  10.883  .99580 

45 

16 

.09179  .09218  10.848  .99578 

44 

17 

.09208  .09247  10.814  .99575 

43 

18 

.09237  .09277  10.780  .99572 

42 

19 

.09  266  .09  306  10.746  .99  570 

41 

20 

.09295  .09335  10.712  .99567 

40 

21 

.09324  .09365  10.678  .99564 

39 

22 

.09353  .09394  10.645  .99562 

38 

23 

.09382  .09423  10.612  .99559 

37 

24 

.09411  .09453  10.579  .99556 

36 

25 

.09440  .09482  10.546  .99553 

35 

26 

.09469  .09511  10.514  .99551 

34 

27 

.09498  .09541  10.481  .99548 

33 

28 

.09527  .09570  10.449  .99545 

32 

29 

.09556  .09600  10.417  .99542 

31 

30 

.09585  .09629  10.385  .99540 

30 

31 

.09614  .09658  10.354  .99537 

29 

32 

.09642  .09688  10.322  .99534 

28 

33 

.09671  .09717  10.291  .99531 

27 

34 

.09  700  .09  746  10.260  .99  528 

26 

35 

.09  729  .09  776  10.229  .99  526 

25 

36 

.09758  .09805  10.199  .99523 

24 

37 

.09  787  .09  834  10.168  .99  520 

23 

38 

.09816  .09864  10.138  .99517 

22 

39 

.09845  .09893  10.108  .99514 

21 

40 

.09874  .09923  10.078  .99511 

20 

41 

.09903  .09952  10.048  .99508 

19 

42 

.09932  .09981  10.019  .99506 

18 

43 

.09961  .10011  9.9893  .99503 

17 

44 

.09990  .10040  9.9601  .99500 

16 

45 

.10019  .10069  9.9310  .99497 

15 

46 

.10048  .10099  9.9021  .99494 

14 

47 

.10077  .10128  9.8734  .99491 

13 

48 

.10106  .10158  9.8448  .99488 

12 

49 

.10  135  .10  187  9.8164  .99  485 

11 

50 

.10164  .10216  9.7882  .99482 

1O 

51 

.10  192  .10  246  9.7601  .99  479 

9 

52 

.10221  .10275  9.7322  .99476 

8 

53 

.10250  .10305  9.7044  .99473 

7 

54 

.10279  .10334  9.6768  .99470 

6 

55 

.10308  .10363  9.6493  .99467 

5 

56 

.10337  .10393  9.6220  .99464 

4 

57 

.10366  .10422  9.5949  .99461 

3 

58 

.10395  .10452  9.5679  .99458 

2 

59 

.10424  .10481  9.5411  .99455 

1 

6O 

.10453  .10510  9.5144  .99452 

O 

/ 

cos        cot        tan        sin 

/ 

84° 

NATURAL   FUNCTIONS 


61 


6D 

/ 

sin    tan    cot    cos 

/ 

o 

.10453  .10510  9.5144  .99452 

GO 

1 

.10482  JO  540  9.4878  .99449 

59 

2 

.10511  .10569  9.46L4  .99446 

58 

3 

.10540  .10599  9.4352  .99443 

57 

4 

.10569  .10628  9.4090  .99440 

56 

5 

.10597  .10657  9.3S31  .99437 

55 

6 

.10626  .10687  9.3572  .99434 

54 

7 

.106S5  .10716  93315  .99431 

53 

8 

.10681  .10746  9.3060  .99428 

52 

9  .10713  .10775  9.2806  .99-124 

51 

10 

.10742  .10805  9.2553  .99421 

50 

11 

.10771  .10834  9.2302  .99418 

49 

12 

.10800  .10863  9.2052  .99415 

48 

13 

.10829  .10893  9.1803  .99412 

47 

14 

.10858  .10922  9.1555  .99409 

46 

15 

.10837  .10952  9.1309  .99406 

45 

16 

.10916  .10981  9.1065  .99402 

44 

17 

.10945  .11011  9.0821  .99399 

43 

18 

.10973  .11040  9.0579  .99396 

42 

19 

.11002  .11070  9.0338  .99393 

41 

20 

.11031  .11099  9.0098  .99390 

4O 

21 

.11050  .11  128  8.9860  .99386 

39 

22 

.11089  .11  158  8.9623  .99383 

38 

23 

.11  118  .11  187  8.9387  .99380 

37 

24 

.11  147  .11217  8.9152  .99377 

36 

25 

.11176  .11246  8.8919  .99374 

35 

26 

.11205  .11276  8.8686  .99370 

34 

27 

.11234  .11305  8.8455  .99367 

33 

28 

.11263  .11335  8.8225  .99364 

32 

29 

.11291  .11364  8.7996  .99360 

31 

30 

.11  320  .11394  8.7769  .99357 

30 

31 

.11349  .11423  8.7542  .99354 

29 

32 

.11378  .11452  8.7317  .99351 

28 

33 

.11407  .11482  8.7093  .99347 

27 

34 

.11436  .11511  8.6870  .99344 

26 

35 

.11465  .11541  8.6648  .99341 

25 

36 

.11494  .11570  8.6427  .99337 

24 

37 

.11523  .11600  8.6208  .99334 

23 

38 

.11552  .11629  8.5989  .99331 

22 

39 

.11580  .11659  8.5772  .99327 

21 

4O 

.11609  .11688  8.5555  .99324 

2O 

41 

.11  638  .11  718  8.5340  .99320 

19 

42 

.11667  .11  747  8.5126  .99317 

18 

43 

.11696  .11  777  8.4913  .99314 

17 

44 

.11725  .11806  8.4701  .99310 

16 

45 

.11  754  .11  836  8.4490  .99307 

15 

46 

.11  783  .11865  8.4280  .99303 

14 

47 

'.11812  .11895  8.4071  .99300 

13 

48 

.11840  .11924  8.3863  .99297 

12 

49 

.11869  .11954  8.3656  .99293 

11 

50 

.11898  .11983  8.3450  .99290 

1O 

51 

.11927  .12013  8.3245  .99286 

9 

52 

.11956  .12042  8.3041  .99283 

8 

53 

.11985  .12072  8.2838  .99279 

7 

54 

.12014  .12101  8.2636  .99276 

6 

55 

.12043  .12131  8.2434  .99272 

5 

56 

.12071  .12160  8.2234  .99269 

4 

57 

.12  100  .12  190  8.2035  .99265 

3 

58 

.12  129  .12  219  8.1837  .99  262 

2 

59 

.12  158  .12249  8.1640  .99258 

1 

6O 

.12187  .12278  8.1443  .99255 

O 

/ 

cos    cot   tan    sin 

/ 

83° 

7° 

/ 

sin    tan    cot    cos 

/ 

0 

.12187  .12278  8.1443  .99255 

60 

1 

.12216  .12308  8.1248  .99251 

59 

2 

.12245  .12338  8.1054  .99248 

58 

3 

.12274  .12367  8.0860  .99244 

57 

4 

.12302  .12397  8.0667  .99240 

56 

5 

.12331  .12426  8.0476  .99237 

55 

6 

.12360  .12456  80285  .99233 

54 

•  7 

.12389  .12485  8.C095  .99230 

53 

8 

.12418  .12515  7.9906  .99226 

52 

9 

.12447  .12544  7.9718  .99222 

51 

1O 

.12476  .12574  7.9530  .99219 

50 

11 

.12501  .12603  7.9344  .99215 

49 

12 

.12533  .12633  7.9158  .99211 

48 

13 

.12562  .12662  7.8973  .99208 

47 

14 

.12591  .12692  7.8789  .99204 

46 

15 

.12  620  .12  722  7.8606  .99  200 

45 

16 

.12649  .12751  7.8424  .99197 

44 

17 

.12678  .12781  7.8243  .99193 

43 

18 

.12  706  .12  810  7.8062  .99  189 

42 

19 

.12735  .12840  7.7882  .99186 

41 

2O 

.12764  .12869  7.7704  .99182 

4O 

21 

.12793  .12899  7.7525  .99178 

39 

22 

.12822  .12929  7.7348  .99175 

38 

23 

.12851  .12958  7.7171  .99171 

37 

24 

.12880  .12988  7.6996  .99167 

36 

25 

.12908  .13017  7.6821  .99163 

35 

26 

.12937  .13047  7.6647  .99160 

34 

27 

.12966  .13076  7.6473  .99156 

33 

28 

.12  995  .13  106  7.6301  .99  152 

32 

29 

.13024  .13136  7.6129  .99148 

31 

30 

.13053  .13165  7.5958  .99144 

30 

31 

.13081  .13  195  7.5787  .99141 

29 

32 

.13110  .13224  7.5618  .99137 

28 

33 

.13  139  .13  254  7.5449  .99  133 

27 

34 

.13  168  .13  284  7.5281  .99  129 

26 

35 

.13197  .13313  7.5113  .99125 

25 

36 

.13226  .13343  7.4947  .99122 

24 

37 

.13254  .13372  7.4781  .99118 

23 

38 

.13283  .13402  7.4615  .99114 

22 

39 

.13  312  .13  432  7.4451  .99  110 

21 

4O 

.13341  .13461  7.4287  .99106 

20 

41 

.13370  .13491  7.4124  .99102 

19 

42 

.13  399  .13  521  7.3962  .99  098 

18 

43 

.13427  .13550  7.3300  .99094 

17 

44 

.13456  .13580  7.3639  .99091 

16 

45 

.13485  .13609  7.3479  .99087 

15 

46 

.13514  .13639  7.3319  .99083 

14 

47 

.13543  .13669  7.3160  .99079 

13 

48 

.13  572  .13  698  7.3002  .99  075 

12 

49 

.13  600  .13  728  7.2844  .99  071 

11 

50 

.13629  .13758  7.2687  .99067 

1O 

51 

.13658  .13  787  7.2531  .99063 

9 

52 

.13687  .13817  7.2375  .99059 

8 

53 

.13716  .13846  7.2220  .99055 

7 

54 

.13744  .13876  7.2066  .99051 

6 

55 

.13773  .13906  7.1912  .99047 

5 

56 

.13802  .13935  7.1759  .99043 

4 

57 

.13831  .13965  7.1607  .99039 

3 

58 

.13860  .13995  7.1455  .99035 

2 

59 

.13  889  .14  024  7.1304  .99  031 

1 

60 

.13917  .14054  7.1154  .99027 

O 

/ 

cos    cot   tan    sin 

/ 

82° 

62 


NATURAL  FUNCTIONS 


8° 

/ 

sin        tan        cot        cos 

/ 

o 

.13917  .14054  7.1154  .99027 

6O 

1 

.13946  .14084  7.1004  .99023 

59 

2 

13975  .14113  7.0855  .99019 

58 

3 

.14004  .14143  7.0706  .99015 

57 

4 

.14033  .14173  7.0558  .99011 

56 

5 

.14061  .14202  7.0410  .99006 

55 

6 

.14  090  .14  232  7.0264  .99  002 

54 

7 

.14119  .14262  7.0117  .98998 

53 

8 

.14  148  .14291  6.9972  .98994 

52 

9 

.14177  .14321  6.9827  .98990 

51 

1O 

.14205  .14351  6.9682  .98986 

5O 

11 

.14234  .14381  6.9538  .98982 

49 

12 

.14263  .14410  6.9395  .98978 

48 

13 

.14  292  .14  440  6.9252  .98  973 

47 

14 

.14320  .14470  6.9110  .98969 

46 

15 

.14349  .14499  6.8969  .98965 

45 

16 

.14  378  .14  529  6.8828  .98  961 

44 

17 

.14407  .14559  6.8687  .98957 

43 

18 

.14  436  .14  588  6.8548  .98  953 

42 

19 

.14464  .14618  6.8408  .98948 

41 

2O 

.14493  .14648  6.8269  .98944 

4O 

21 

.14522  .14678  6.8131  .98940 

39 

22 

.14551  .14707  6.7994  .98936 

38 

23 

.14580  .14737  6.7856  .98931 

37 

24 

.14  608  .14  767  6.7720  .98  927 

36 

25 

.14637  .14796  6.7584  .98923 

35 

26 

.14666  .14826  6.7448  .98919 

34 

27 

.14695  .14856  6.7313  .98914 

33 

28 

.14723  .14886  6.7179  .98910 

32 

29 

.14752  .14915  6.7045  .98906 

31 

30 

.14781  .14945  6.6912  .98902 

30 

31 

.14810  .14975  6.6779  .98897 

29 

32 

.14  838  .15  005  6.6646  .98  893 

28 

33 

.14867  .15034  6.6514  .98889 

27 

34 

.14  896  .15  064  6.6383  .98  884 

26 

35 

.14925  .15094  6.6252  .98880 

25 

36 

.14954  .15  124  6.6122  .98876 

24 

37 

.14  982  .15  153  6.5992  .98  871 

23 

38 

.15011  .15  183  6.5863  .98867 

22 

39 

.15  040  .15  213  6.5734  .98  863 

21 

40 

.15  069  .15  243  6.5606  .98  858 

20 

41 

.15097  .15272  6.5478  .98854 

19 

42 

.15  126  .15  302  6.5350  .98  849 

18 

43 

.15  155  .15  332  6.5223  .98  845 

17 

44 

.15  184  .15  362  6.5097  .98  841 

16 

45 

.15212  .15391  6.4971  .98836 

15 

46 

.15  241  .15  421  6.4846  .98  832 

14 

47 

.15270  .15451  6.4721  .98827 

13 

48 

.15  299  .15  481  6.4596  .98  823 

12 

49 

.15327  .15511  6.4472  .98818 

11 

50 

.15  356  .15  540  6.4348  .98  814 

10 

51 

.15385  .15570  6.4225  .98809 

9 

52 

.15  414  .15  600  6.4103  .98  805 

8 

53 

.15442  .15630  6.3980  .98800 

7 

54 

.15471  .]5660  6.3859  .98796 

6 

55 

.15  500  .15  689  6.3737  .98  791 

5 

56 

.15  529  .15  719  6.3617  .98  787 

4 

57 

.15557  .15749  6.3496  .98782 

3 

58 

.15  586  .15  779  6.3376  .98  778 

2 

59 

.15  615  .15  809  6.3257  .98  773 

1 

6O 

.15  643  .15  838  6.3138  .98  769 

O 

/ 

cos        cot        tan        sin 

/ 

81° 

9° 

/ 

sin        tan        cot        cos 

/ 

O 

.15643  .15838  6.3138  .98769 

60^ 

1 

.15  672  .15  868  6.3019  .98  764 

59 

2 

.15  701  .15898  6.2901  .98760 

58 

3 

.15  730  .15928  6.2783  .98755 

57 

4 

.15  758  .15958  6.2666  .98751 

56 

5 

.15  787  .15  988  6.2549  .98  746 

55 

6 

.15816  .16017  6.2432  .98741 

54 

7 

.15845  .16047  6.2316  .98737 

53 

8 

.15873  .16077  6.2200  .98732 

52 

9 

.15  902  .16  107  6.2085  .98  728 

51 

10 

.15931  .16137  6.1970  .98723 

50 

11 

.15  959  .16  167  6.1856  .98  718 

49 

12 

.15988  .16196  6.1742  .98714 

48 

13 

.16017  .16226  6.1628  .98709 

47 

14 

.16046  .16256  6.1515  .98704 

46 

15 

.16074  .16286  6.1402  .98700 

45 

16 

.16103  .16316  6.1290  .98695 

44 

17 

.16132  .16346  6.1178  .98690 

43 

18 

.16  160  .16  376  6.1066  .98  686 

42 

19 

.16189  .16405  6.0955  .98681 

41 

20 

.16218  .16435  6.0844  .98676 

40 

21 

.16246  .16465  6.0734  .98671 

39 

22 

.16275  .16495  6.0624  .98667 

38 

23 

.16304  .16525  6.0514  .98662 

37 

24 

.16333  .16555  6.0405  .98657 

36 

25 

.16361  .16585-6.0296  .98652 

35 

26 

.16390  .16615  6.0188  .98648 

34 

27 

.16419  .16645  6.0080  .98643 

33 

28 

.16447  .16674  5.9972  .98638 

32 

29 

.16476  .16704  5.9865  .98633 

31 

30 

.16505  .16734  5.9758  .98629 

30 

31 

.16533  .16764  5.9651  .98624 

29 

32 

.16562  .16794  5.9545  .98619 

28 

33 

.16591  .16824  5.9439  .98614 

27 

34 

.16620  .16854  5.9333  .98609 

26 

35 

.16648  .16884  5.9228  .98604 

25 

36 

.16677  .16914  5.9124  .98600 

24 

37 

.16706  .16944  5.9019  .98595 

23 

38 

.16734  .16974  5.8915  .98590 

22 

39 

.16763  .17004  5.8811  .98585 

21 

40 

.16792  .17033  5.8708  .98580 

2O 

41 

..16820  .17063  5.8605  .98575 

19 

42 

.16849  .17093  5.8502  .98570 

18 

43 

.16878  .17123  5.8400  .98565 

17 

44 

.16906  .17153  5.8298  .98561 

16 

45 

.16935  .17183  5.8197  .98556 

15 

46 

.16964  .17213  5.8095  .98551 

14 

47 

.16992  .17243  5.7994  .98546 

13 

48 

.17021  .17273  5.7894  .98541 

12 

49 

.17050  .17303  5.7794  .98536 

11 

5O 

.17078  .17333  5.7694  .98531 

10 

51 

.17107  .17363  5.7594  .98526 

9 

52 

.17136  .17393  5.7495  .98521 

8 

53 

.17164  .17423  5.7396  .98516 

7 

54 

.17193  .17453  5.7297  .98511 

6 

55 

.17222  .17483  5.7199  .98506 

5 

56 

.17250  .17513  5.7101  .98501 

4 

57 

.17279  .17543  5.7004  .98496 

3 

58 

.17308  .17573  5.6906  .98491 

2 

59 

.17336  .17603  5.6809  .98486 

1 

60 

.17365  .17633  5.6713  .98481 

O 

/ 

cos        cot        tan        sin 

/ 

80° 

NATURAL   FUNCTIONS 


63 


10° 

/ 

sin    tan    cot    cos 

/ 

o 

.17365  .17633  5.6713  .98481 

6O 

1 

.17393  .17663  5.6617  .98476 

59 

2 

.17422  .17693  5.6521  .98471 

58 

3 

.17451  .17723  5.6425  .98466 

57 

4 

.17479  .17753  5.6329  .98461 

56 

5 

.17  508  .17  783  5.6234  .98  455 

55 

6 

.17537  .17813  5.6140  .98450 

54 

7 

.17565  .17843  5.6045  .98445 

53 

8 

.17594  .17873  5.5951  .98440 

52 

9 

.17623  .17903  5.5857  .98435 

51 

1O 

.17651  .17933  5.5764  .98430 

50 

11 

.17680  .17963  5.5671  .98425 

49 

12 

.17  708  .17  993  5.5578  .98  420 

48 

13 

.17737  .18023  5.5485  .98414 

47 

14 

.17766  .18053  5.5393  .98409 

46 

15 

.17794  .18083  5.5301  .98404 

45 

16 

.17823  .18113  5.5209  .98399 

44 

17 

.17852  .18143  5.5118  .98394 

43 

18 

.17880  .18173  5.5026  .98389 

42 

19 

.17909  .18203  5.4936  .98383 

41 

20 

.17937  .18233  5.4845  .98378 

40 

21 

.17  966  .18  263  5.4755  .98  373 

39 

22 

.17  995  .18  293  5.4665  .98  368 

38 

23 

.18023  .18323  5.4575  .98362 

37 

24 

.18052  .18353  5.4486  .98357 

36 

25 

.18081  .18384  5.4397  .98352 

35 

26 

.18  109  .18  414  5.4308  .98  347 

34 

27 

.18  138  .18  444  5.4219  .98  341 

33 

28 

.18166  .18474  5.4131  .98336 

32 

29 

.18195  .18504  5.4043  .98331 

31 

'30 

.18  224  .18  534  5.3955  .98  325 

3O 

31 

.18  252  .18  564  5.3868  .98  320 

29 

32 

.18281  .18594  5.3781  .98315 

28 

33 

.18309  .18624  5.3694  .98310 

27 

34 

.18  338  .18  654  5.3607  .98  304 

26 

35 

.18367  .18684  5.3521  .98299 

25 

36 

.18395  .18714  5.3435  .98294 

24 

37 

.18  424  .18  745  5.3349  .98  288 

23 

38 

.18452  .18775  5.3263  .98283 

22 

39 

.18481  .18805  5.3178  .98277 

21 

40 

.18  509-  .18  835  5.3093  .98272 

2O 

41 

.18  538  .18  865  5.3008  .98  267 

19 

42 

.18  567  .18  895  5.2924  .98  261 

18 

43 

.18  595  .18  925  5.2839  .98  256 

17 

44 

.18624  .18955  5.2755  .98250 

16 

45 

.18652  .18986  5.2672  .98245 

15 

46 

.18681  .19016  5.2588  .98240 

14 

47 

.13710  .19046  5.2505  .98234 

13 

48 

.18738  .19076  5.2422  .98229 

12 

49 

.18  767  .19  106  5.2339  .98  223 

11 

50 

.18  795  .19  136  5.2257  .98  218 

1O 

51 

.18  824  .19  166  5.2174  .98  212 

9 

52 

.18  852  .19  197  5.2092  .98  207 

8 

53 

.18881  .19227  5.2011  .98201 

7 

54 

.18910  .19257  5.1929  .98196 

6 

55 

.18938  .19287  5.1848  .98190 

5 

56 

.18967  .19317  5.1767  .98185 

4 

57 

.18995  .19347  5.1686  .98179 

3 

58 

.19024  .19378  5.1606  .98174 

2 

59 

.19052  .19408  5.1526  .98168 

1 

60 

.19  081  .19  438  5.1446  .98  163 

0 

/ 

cos    cot   tan    sin 

/ 

79° 

11° 

/ 

sin   tan   cot   cos 

/ 

0 

.19081  .19438  5.1446  .98163 

60 

1 

.19109  .19468  5.1366  .98157 

59 

2 

.19138  .19498  5.1286  .98152 

58 

3 

.19  167  .19  529  5.1207  .98  146 

57 

4 

.19195  .19559  5.1128  .98140 

56 

5 

.19  224  .19  589  5.1049  .98  135 

55 

6 

.19  252  .19  619  5.0970  .98  129 

54 

7 

.19281  .19649  5.0892  .98124 

53 

8 

.19309  .19680  5.0814  .98118 

52 

9 

.19  338  .19  710  5.0736  .98  112 

51 

1O 

.19  366  .19  740  5.0658  .98  107 

5O 

11 

.19395  .19770  5.0581  .98101 

49 

12 

.19423  .19801  5.0504  .98096 

48 

13 

.19452  .19831  5.0427  .98090 

47 

14 

.19481  .19861  5.0350  .98084 

46 

15 

.19  509  .19  891  5.0273  .98  079 

45 

16 

.19538  .19921  5.0197  .98073 

44 

17 

.19566  .19952  5.0121  .98067 

43 

18 

.19595  .19982  5.0045  .98061 

42 

19 

.19  623  .20  012  4.9969  .98  056 

41 

2O 

.19652  .20042  4.9894  .98050 

4O 

21 

.19680  .20073  4.9819  .98044 

39 

22 

.19709  .20103  4.9744  .98039 

38 

23 

.19  737  .20  133  4.9669  .98  033 

37 

24 

.19766  .20164  4.9594  .98027 

36 

25 

.19794  .20194  4.9520  .98021 

35 

26 

.19  823  .20  224  4.9446  .98  016 

34 

27 

.19851  .20254  4.9372  .98010 

33 

28 

.19  880  .20  285  4.9298  .98  004 

32 

29 

.19908  .20315  4.9225  .97998 

31 

30 

.19937  .20345  4.9152  .97992 

3O 

31 

.19  965  .20  376  4.9078  .97  987 

29 

32 

.19994  .20406  4.9006  .97981 

28 

33 

.20022  .20436  4.8933  .97975 

27 

34 

.20051  .20466  4.8860  .97969 

26 

35 

.20079  .20497  4.8788  .97963 

25 

36 

.20  108  .20  527  4.8716  .97  958 

24 

37 

.20  136  .20  557  4.8644  .97  952 

23 

38 

.20  165  .20  588  4.8573  .97  946 

22 

39 

.20  193  .20  618  4.8501  .97  940 

21 

4O 

.2022?  .20648  4.8430  .97934 

2O 

41 

.20  250  .20  679  4.8359  .97  928 

19 

42 

.20  279  .20  709  4.8288  .97  922 

18 

43 

.20307  .20739  4.8218  .97916 

17 

44 

.20  336  .20  770  4.8147  .97  910 

16 

45 

.20364  .20800  4.8077  .97905 

15 

46 

.20393  .20830  4.8007  .97899 

14 

47 

.20421  .20861  4.7937  .97893 

13 

48 

.20450  .20891  4.7867  .97887 

12 

49 

.20478  .20921  4.7798  .97881 

11 

5O 

.20  507  .20  952  4.7729  .97  875 

1O 

51 

.20  535  .20  982  4.7659  .97  869 

9 

52 

.20  563  .21  013  4.7591  .97  863 

8 

53 

.20  592  .21  043  4.7522  .97  857 

7 

54 

.20  620  .21  073  4.7453  .97  851 

6 

55 

.20649  .21  104  4.7385  .97845 

5 

56 

.20677  .21  134  4.7317  .97839 

4 

57 

.20706  .21  164  4.7249  .97833 

3 

58 

.20  734  .21  195  4.7181  .97  827 

2 

59 

.20763  .21225  4.7114  .97821 

1 

60 

.20  791  .21  256  4.7046  .97  815 

O 

/ 

cos    cot   tan    sin 

/ 

78° 

64 


NATURAL   FUNCTIONS 


12° 

/ 

sin   tan   cot   cos 

/ 

o 

.20  791  .21  256  4.7046  .97  815 

60^ 

1 

.20  820  .21  286  4.6979  .97  809 

59 

2 

.20848  .21316  4.6912  .97803 

58 

3 

.20  877  .21  347  4.6845  .97  797 

57 

4 

.20905  .21377  4.6779  .97791 

56 

5 

.20  933  .21  408  4.6712  .97  784 

55 

6 

.20  962  .21  438  4.6646  .97  778 

54 

7 

.20  990  .21  469  4.6580  .97  772 

53 

8 

.21  019  .21  499  4.6514  .97  766 

52 

9 

.21  047  .21  529  4.6448  .97  760 

51 

1O 

.21  076  .21  560  4.6382  .97  754 

5O 

11 

.21  104  .21  590  4.6317  .97  748 

49 

.12 

.21  132  .21  621  4.6252  .97  742 

48 

13 

.21  161  .21  651  4.6187  .97  735 

47 

14 

.21  189  .21  682  4.6122  .97  729 

46 

15 

.21  218  .21  712  4.6057  .97  723 

45 

16 

.21  246  .21  743  4.5993  .97  717 

44 

17 

.21  275  .21  773  4.5928  .97  711 

43 

18 

.21  303  .21  804  4.5864  .97  705 

42 

19 

.21331  .21834  4.5800  .97698 

41 

20 

.21  360  .21  864  4.5736  .97  692 

4O 

21 

.21  388  .21  895  4.5673  .97  686 

39 

22 

.21  417  .21  925  4.5609  .97  680 

38 

23 

.21445  .21956  4.5546  .97673 

37 

24 

.21  474  .21  986  4.5483  .97  667 

36 

25 

.21502  .22017  4.5420  .97661 

35 

26 

.21  530  .22  047  4.5357  .97  655 

34 

27 

.21  559  .22  078  4.5294  .97  648 

33 

28 

.21  587  .22  108  4.5232  .97  642 

32 

29 

.21616  .22139  4.5169  .97636 

31 

30 

.21644  .22  169  4.5107  .97630 

30 

31 

.21  "57^.22  200  4.5045  .97623 

29 

32 

.21701  .22231  4.4983  .97617 

28 

33 

.21  729  .22261  4.4922  .97611 

27 

34 

.21  758  .22  292  4.4860  .97  604 

26 

35 

.21  786  .22  322  4.4799  .97  598 

25 

36 

.21  814  .22  353  4.4737  .97  592 

24 

37 

.21  843  .22  383  4.4676  .97  585 

23 

38 

.21871  .22414  4.4615  .97579 

22 

39 

.21  899  .22  444  4.4555  .97  573 

21 

4O 

.21928  .22475  4.4494  .97566 

20 

41 

.21  956  .22  505  4.4434  .97  560 

19 

42 

.21  985  .22  536  4.4373  .97  553 

18 

43 

.22013  .22567  4.4313  .97547 

17 

44 

.22  041  .22  597  4.4253  .97  541 

16 

45 

.22  070  .22  628  4.4194  .97  534 

15 

46 

.22  098  .22  658  4.4134  .97  528 

14 

47 

.22  126  .22  689  4.4075  .97  521 

13 

48 

.22155  .22719  4.4015  .97515 

12 

49 

.22  183  .22  750  4.3956  .97  508 

11 

50 

.22  212  .22  781  4.3897  .97  502 

1O 

51 

.22240  .22811  4.3838  .97496 

9 

52 

.22  268  .22  842  4.3779  .97  489 

8 

53 

.22297  .22872  4.3721  .97483 

7 

54 

.22'  325  .22903  4.3662  .97476 

6 

55 

.22  353  .22  934  4.3604  .97  470 

5 

56 

.22  382  .22  964  4.3546  .97  463 

4 

57 

.22410  .22995  4.3488  .97457 

3 

58 

.22  438  .23  026  4.3430  .97  450 

2 

59 

.22  467  .23  056  4.3372  .97  444 

1 

6O 

.22  495  .23  087  4.3315  .97  437 

0 

/ 

cos   cot   tan   sin 

/ 

77° 

13° 

/ 

sin   tan   cot   cos 

/ 

O 

.22495  .23087  4.3315  .97437 

60 

1 

.22523  .23117  4.3257  .97430 

59 

2 

.22  552  .23  148  4.3200  .97  424 

58 

3 

.22580  .23179  4.3143  .97417 

57 

4 

.22  608  .23  209  4.3086  .97  411 

56 

5 

.22  637  .23  240  4.3029  .97  404 

55 

6 

.22  665  .23  271  4.2972  .97  398 

54 

7 

.22  693  .23  301  4.2916  .97  391 

53 

8 

.22  722  .23  332  4.2859  .97  384 

52 

9 

.22  750  .23  363  4.2803  .97  378 

51 

10 

.22  778  .23  393  4.2747  .97  371 

5O 

11 

.22  807  .23  424  4.2691  .97  365 

49 

12 

.22  835  .23  455  4.2635.  .97  358 

48 

13 

.22863  .23485  4.2580  .97351 

47 

14 

.22892  .23  516  4.2524  .97345 

46 

-  15 

.22  920  .23  547  4.2468  .97  338 

45 

16 

.22948  .23578  4.2413  .97331 

44 

17 

.22  977  .23  608  4.2358  .97  325 

43 

18 

.23  005  .23  639  4.2303  .97  318 

42 

19 

.23033  .23670  4.2248  .97311 

41 

2O 

.23  062  .23  700  4.2193  .97  304 

4O 

21 

.23  090  .23  731  4.2139  .97  298 

39 

22 

.23  118  .23  762  4.2084  .97291 

38 

23 

.23  146  .23  793  4.2030  .97  284 

37 

21 

.23  175  .23  823  4.1976  .97  278 

36 

25 

.23  203  .23  854  4.1922  .97  271 

35 

26 

.23  231  .23  885  4.1868  .97  264 

34 

27 

.23  260  .23  916  4.1814  .97  257 

33 

28 

.23  288  .23  946  4.1760  .97  251 

32 

29 

.23  316  .23  977  4.1706  .97  244 

31 

3O 

.23345  .24008  4.1653  .97237 

3O 

31 

.23373  .24039  4.1600  .97230 

29 

32 

.23  401  .24  069  4.1547  .97  223 

28 

33 

.23  429  .24  100  4.1493  .97  217 

27 

34 

.23  458  .24  131  4.1441  .97  210 

26 

35 

.23  486  .24  162  4.1388  .97  203 

25 

36 

.23  514  .24  193  4.1335  .97  196 

24 

37 

.23  542  .24  223  4.1282  .97  189 

23 

38 

.23  571  .24  254  4.1230  .97  182 

22 

39 

.23  599  .24  285  4.1178  .97  176 

21 

4O 

.23627  .24316  4.1126  .97169 

2O 

41 

.23  656  .24  347  4.1074  .97  162 

19 

42 

.23  684  .24  377  4.1022  .97  155 

18 

43 

.23  712  .24  408  4.0970  .97  148 

17 

44 

.23  740  .24  439  4.0918  .97  141 

16 

45 

.23  769  .24  470  4.0867  .97  134 

15 

46 

.23797  .24  501  4.0815  .97127 

14 

47 

.23  825  .24  532  4.0764  .97  120 

13 

48 

.23  853  .24  562  4.0713  .97  113 

12 

49 

.23  882  .24  593  4.0662  .97  106 

11 

50 

.23910  .24624  4.0611  .97100 

10 

51 

.23  938  .24  655  4.0560  .97  093 

9 

52 

.23  966  .24  686  4.0509  .97  086 

8 

53 

.23  995  .24  717  4.0459  .97  079 

7 

54 

.24  023  .24  747  4.0408  .97  072 

6 

55 

.24051  .24778  4.0358  .97065 

5 

56 

.24  079  .24  809  4.0308  .97  058 

4 

57 

.24  108  .24  840  4.0257  .97  051 

3 

58 

.24  136  .24  871  4.0207  .97  044 

2 

59 

.24  164  .24  902  4.0158  .97  037 

1 

60 

.24192  .24933  4.0108  .97030 

O 

/ 

cos   cot   tan   sin 

/ 

76° 

NATURAL   FUNCTIONS 


65 


14° 

/ 

sin    tan    cot    cos 

/ 

o 

.24  192  .24  933  4.0108  .97  030 

6O 

1 

.24  220  .24  964  4.0058  .97  023 

59 

2 

.24249  .24995  4.0009  .97015 

58 

3 

.24  277  .25  026  3.9959  .97  008 

57 

4 

.24  305  .25  056  3.9910  .97  001 

56 

5 

.24  333  .25  087  3.9861  .96  994 

55 

6 

.24362  .25  118  3.9812  .96987 

54 

7 

.24  390  .25  149  3.9763  .96  980 

53 

8 

.24  418  .25  180  3.9714  .96973 

52 

9 

.24446  .25211  3.9665  .96966 

51 

10 

.24  474  .25  242  3.9617  .96  959 

5O 

11 

.24  503  .25  273  3.9568  .96  952 

49 

12 

.24  531  .25  304  3  9520  .96  945 

48 

13 

.24559  .25335  3.9471  .96937 

47 

14 

.24587  .25*366  3.9423  .96930 

46 

15 

.24  615  .25  397  3.9375  .96  923 

45 

16 

.24644  .25428  3.9327  .96916 

44 

17. 

.24  672  .25  459  3.9279  .96  909 

43 

18 

.24  700  .25  490  3.9232  .96  902 

42 

19 

.24  728  .25  521  3.9184  .96  894 

41 

2O 

.24756  .25552  3.9136  .96887 

4O 

21 

.24  784  .25  583  3.9089  .96  880 

39 

22 

.24  813  .25  614  3.9042  .96  873 

38 

23 

.24841  .25645  3.8995  .96866 

37 

24 

.24  869  .25  676  3.8947  .96  858 

36 

25 

.24897  .25707  3.8900  .96851 

35 

26 

.24  925  .25  738  3.8854  .96  844 

34 

27 

.24  954  .25  769  3.8807  .96  837 

33 

|  28 

.24982  .25800  3.8760  .96829 

32 

29 

.25010  .25831  3.8714  .96822 

31 

3D 

.25038  .25862  3.8667  .96815 

30 

31 

.25066  .25893  3.8621  .96807 

29 

32 

.25094  .25924  3.8575  .96800 

28 

33 

.25  122  .25  955  3.8528  .96  793 

27 

34 

.25  151  .25  986  3.8482  .96  786 

26 

35 

.25179  .26017  3.8436  .96778 

25 

36 

.25  207  .26  048  3.8391  .96  771 

24 

37 

.25  235  .26  079  3.8345  .96  764 

23 

38 

.25  263  .26  110  3.8299  .96  756 

22 

39 

.25  291  .26  141  3.8254  .96  749 

21 

4O 

.25  320  .26  172  3.8208  .96  742 

2O 

41 

.25348  .26203  3.8163  .96734 

19 

42 

.25376  .26235  3.8118  .96727 

18 

43 

.25  404  .26  266  3.8073  .96  719 

17 

44 

.25432  .26297  3.8028  .96712 

16 

45 

.25  460  .26  328  3.7983  .96  705 

15 

46 

.25  488  .26  359  3.7938  .96  697 

14 

47 

.25516  .26390  3.7893  .96690 

13 

48 

.25  545  .26  421  3.7848  .96  682 

12 

49 

.25573  .26452  3.7804  .96675 

11 

50 

.25  601  .26  483  3.7760  .96  667 

10 

51 

.25629  .26515  3.7715  .96660 

9 

52 

.25657  .26546  3.7671  .96653 

8 

53 

.25685  .26577  3.7627  .96645 

7 

54 

.25713  .26608  3.7583  .96638 

6 

55 

.25741  .26639  3.7539  .96630 

5 

56 

.25  769  .26  670  3.7495  .96  623 

4 

57 

.25798  .26701  3.7451  .96615 

3 

58 

.25  826  .26  733  3.7408  .96  608 

2 

59 

.25  854  .26  764  3.7364  .96  600 

1 

60 

.25882  .26795  3.7321  .96593 

O 

/ 

cos    cot    tan    sin 

/ 

75° 

15° 

/ 

sin    tan    cot    cos 

/ 

O 

.25  882  .26  795  3.7321  .96  593 

60 

1 

.25  910  .26  826  3.7277  .96  585 

59 

2 

.25  938  .26  857  3.7234  .96  578 

58 

3 

.25  966  .26  888  3.7191  .96  570 

57 

4 

.25  994  .26  920  3.7148  .96  562 

56 

5 

.26022  .26951  3.7105  .96555 

55 

6 

.26050  .26982  3.7062  .96547 

54 

7 

.26079  .27013  3.7019  .96540 

53 

8 

.26  107  .27  044  3.6976  .96  532 

52 

9 

.26  135  .27  076  3.6933  .96  524 

51 

1O 

.26163  .27107  3.6891  .96517 

5O 

11 

.26  191  .27  138  3.6848  .96  509 

49 

12 

.26  219  .27  169  3.6806  .96  502 

48 

13 

.26247  .27201  3.6764  .96494 

47 

14 

.26  275  .27  232  3.6722  .96  486 

46 

15 

.26303  .27263  3.6680  .96479 

45 

16 

.26331  .27294  3.6638  .96471 

44 

17 

.26  359  .27  326  3.6596  .96  463 

43 

18 

.26387  .27357  3.6554  .96456 

42 

19 

.26415  .27388  3.6512  .96448 

41 

2O 

.26443  .27419  3.6470  .96440 

4O 

21 

.26471  .27451  3.6429  .96433 

39 

22 

.26500  .27482  3.6387  .96425 

38 

23 

.26528  .27513  3.6346  .96417 

37 

24 

.26556  .27545  3.6305  .96410 

36 

25 

.26584  .27576  3.6264  .96402 

35 

26 

.26612  .27607  3.6222  .96394 

34 

27 

.26  640  .27  638  3.6181  .96  386 

33 

28 

.26  668  .27  670  3.6140  .96  379 

32 

29 

.26  696  .27  701  3.6100  .96  371 

31 

30 

.26  724  .27  732  3.6059  .96  363 

3O 

31 

.26  752  .27  764  3.6018  .96  355 

29 

32 

.26  780  .27  795  3.5978  .96  347 

28 

33 

.26  808  .27  826  3.5937  .96  340 

27 

34 

.26  836  .27  858  3.5897  .96  332 

26 

35 

.26  864  .27  889  3.5856  .96  324 

25 

36 

.26892  .27921  3.5816  .96316 

24 

37 

.26920  .27952  3.5776  .96308 

23 

38 

.26948  .27983  3.5736  .96301 

22 

39 

.26976  .28015  3.5696  .96293 

21 

4O 

.27004  .28046  3.5656  .96285 

2O 

41 

.27032  .28077  3.5616  .96277 

19 

42 

.27  060  .28  109  3.5576  .96  269 

18 

43 

.27  088  .28  140  3.5536  .96  261 

17 

44 

.27  116  .28  172  3.5497  .96  253 

16 

45 

.27  144  .28  203  3.5457  .96  246 

15 

46 

.27  172  .28  234  3.5418  .96  238 

14 

47 

.27  200  .28  266  3.5379  .96  230 

13 

48 

.27  228  .28  297  3.5339  .96  222 

12 

49 

.27256  .28329  3.5300  .96214 

11 

50 

.27284  .28360  3.5261  .96206 

10 

51 

.27312  .28391  3.5222  .96198 

9 

52 

.27340  .28423  3.5183  .96190 

8 

53 

.27  368  .28  454  3.5144  .96  182 

7 

54 

.27396  .28486  3.5105  .96174 

6 

55 

.27424  .28517  3.5067  .96166 

5 

56 

.27  452  .28  549  3.5028  .96  158 

4 

57 

.27480  .28580  3.4989  .96150 

3 

58 

.27508  .28612  3.4951  .96142 

2 

59 

.27536  .28643  3.4912  .96134 

1 

6O 

.27564  .28675  3.4874  .96126 

O 

/ 

cos    cot   tan    sin 

/ 

74° 

66 


NATURAL   FUNCTIONS 


16° 

/ 

sin    tan    cot    cos 

/ 

|O 

.27  564  .28  675  3.4874  .96  126 

6O 

1 

.27592  .28706  3.4836  .96118 

59 

2 

.27620  .28738  3.4798  .96110 

58 

3 

.27  648  .28  769  3.4760  .96  102 

57 

4 

.27676'.28801  3.4722  .96094 

56 

5 

.27  704  .28  832  3.4684  .96  086 

55 

6 

.27731  .28864  3.4646  .96078 

54 

7 

.27759  .28895  3.4608  .96970 

53 

8 

.27787  .28927  3.4570  .96062 

52 

9 

.27815  .28958  3.4533  .96954 

51 

10 

.27843  .28990  3.4495  .96046 

5O 

11 

.27871  .29021  3.4458  .96037 

49 

12 

.27899  .29053  3.4420  .96029 

48 

13 

.27927  .29084  3.4383  .96021 

47 

14 

.27955  .29116  3.4346  .96013 

46 

15 

.27  983  .29  147  3.4308  .96  005 

45 

16 

.28  Oil  .29  179  3.4271  .95  997 

44 

17 

.28  039  .29  210  3.4234  .95  989 

43 

18 

.28  067  .29  242  3.4197  .95  981 

42 

19 

.28095  .29274  3.4160  .95972 

41 

20 

.28  123  .29  305  3.4124  .95  964 

40 

21 

.28150  .29337  3.4087  .95956 

39 

22 

.28178  .29368  3.4050  .95948 

38 

23 

.28206  .29400  3.4014  .95940 

37 

24 

.28234  .29432  3.3977  .95931 

36 

25 

.28  262  .29  463  3.3941  .95  923 

35 

26 

.28  290  .29  495  3.3904  .95  915 

34 

27 

.28  318  .29  526  3.3868  .95  907 

33 

28 

.28346  .29558  3.3832  .95898 

32 

29 

.28  374  .29  590  3.3796  .95  890 

31 

30 

.28402  .29621  3.3759  .95882 

30 

31 

.28429  .29653  3.3723  .95874 

29 

32 

.28457  .29685  3.3687  .95865 

28 

33 

.28485  .29716  3.3652  .95857 

27 

34 

.28513  .29748  3.3616  .95849 

26 

35 

.28  541  .29  780  3.3580  .95  841 

25 

36 

.28569  .29811  3.3544  .95832 

24 

37 

.28  597  .29  843  3.3509  .95  824 

23 

38 

.28625  .29875  3.3473  .95816 

22 

39 

.28  652  .29  906  3.3438  .95  807 

21 

4O 

.28  680  .29  938  3.3402  .95  799 

20 

41 

.28708  .29970  3.3367  .95791 

19 

42 

.28  736  .30  001  3.3332  .95  782 

18 

43 

.28  764  .30  033  3.3297  .95  774 

17 

44 

.28  792  .30  065  3.3261  .95  766 

16 

45 

.28820  .30097  3.3226  .95757 

15 

46 

.28  847  .30  128  3.3191  .95  749 

14 

47 

.28875  .30160  3.3156  .95740 

13 

48 

.28903  .30192  3.3122  .95732 

12 

49 

.28931  .30224  3.3087  .95724 

11 

50 

.28  959  .30  255  3.3052  .95  715 

1O 

51 

.28  987  .30  287  3.3017  .95  707 

9 

52 

.29015  .30319  3.2983  .95698 

8 

53 

.29042  .30351  3.2948  .95690 

7 

54 

.29  070  .30  382  3.2914  .95  681 

6 

55 

.29  098  .30  414  3.2879  .95  673 

5 

56 

.29  126  .30  446  3.2845  .95  664 

4 

1  57 

.29154  .30478  3.2811  .95656 

3 

58 

.29  182  .30  509  3.2777  .95  647 

2 

59 

.29  209  .30  541  3.2743  .95  639 

1 

6O 

.29237  .30573  3.2709  .95630 

O 

/ 

cos    cot    tan    sin 

/ 

73° 

17° 

/ 

sin    tan    cot    cos 

/ 

O 

.29237  .30573  3.27Q9  .95630 

GO 

1 

.29265  .30605  3.2675  .95622 

59 

2 

.29293  .30637  3.2641  .95613 

58 

3 

.29321  .30669  3.2607  .95605 

57 

4 

.29348  .30700  3.2573  .95596 

56 

5 

.29376  .30732  3.2539  .95588 

55 

6 

.29404  .30764  3.2506  .95579 

54 

7 

.29432  .30796  3.2472  .95571 

53 

8 

.29460  .30828  3.2438  .95562 

52 

9 

.29487  .30860  3.2405  .95554 

51 

1O 

.29515  .30891  3.2371  .95545 

50 

11 

.29543  .30923  3.2338  .95536 

49 

12 

.29571  .30955  3.2305  .95528 

48 

13 

.29599  .30987  3.2272  .95519 

47 

14 

.29626  .31019  3.2238  .95511 

46 

15 

.29654  .31051  3.2205  .95502 

45 

16 

.29682  .31083  3.2172  .95493 

44 

17 

.29710  .31115  3.2139  .95485 

43 

18 

.29  737  .31  147  3.2106  .95  476 

42 

19 

.29765  .31  178  3.2073  .95467 

41 

20 

.29  793  .31  210  3.2041  .95  459 

40 

21 

.29  821  .31  242  3.2008  .95  450 

39 

22 

.29849  .31274  3.1975  .95441 

38 

23 

.29876  .31306  3.1943  .95433 

37 

24 

.29904  .31338  3.1910  .95424 

36 

25 

.29932  .31370  3.1878  .95415 

35 

26 

.29960  .31402  3.1845  .95407 

34 

27 

.29987  .31434  3.1813  .95398 

33 

28 

.30015  .31466  3.1780  .95389 

32 

29 

.30043  .31498  3.1748  .95380 

31 

30 

.30071  .31530  3.1716  .95372 

30 

31 

.30098  .31562  3.1684  .95363 

29 

32 

.30  126  .31  594  3.1652  .95  354 

28 

33 

.30  154  .31  626  3.1620  .95  345 

27 

34 

.30  182  .31  658  3.1588  .95  337 

26 

35 

.30  209  .31  690  3.1556  .95  328 

25 

36 

.30  237  .31  722  3.1524  .95  319 

24 

37 

.30  265  .31  754  3.1492  .95  310 

23 

38 

.30292  .31786  3.1460  .95301 

22 

39 

.30  320  .31  818  3.1429  .95  293 

21 

40 

.30348  .31850  3.1397  .95284 

2O 

41 

.30376  .31882  3.1366  .95275 

19 

42 

.30403  .31914  3.1334  .95266 

18 

43 

.30431  .31946  3.1303  .95257 

17 

44 

.30459  .31978  3.1271  .95248 

16 

45 

.30486  .32010  3.1240  .95240 

15 

46 

.30514  .32042  3.1209  .95231 

14 

47 

.30542  .32074  3.1178  .95222 

13 

48 

.30570  .32106  3.1146  .95213 

12 

49 

.30597  .32139  3.1115  .95204 

11 

50 

-.30625  .32171  3.1084  .95  195 

10 

51 

.30653  .32203  3.1053  .95  186 

9 

52 

.30680  .32235  3.1022  .95  177 

8 

53 

.30  708  .32  267  3.0991  .95  168 

7 

54 

.30  736  .32  299  3.0961  .95  159 

6 

55 

.30763  .32331  3.0930  .95150 

5 

56 

.30  791  .32  363  3.0S99  .95  142 

4 

57 

.30  819  .32  396  3.0868  .95  133 

3 

58 

.30  846  .32  428  3.0838  .95  124 

2 

59 

.30874  .32460  3.0807  .95  115 

1 

6O 

.30902  .32492  3.0777  .95  106 

O 

/ 

cos    cot    tan    sin 

/ 

72° 

NATURAL   FUNCTIONS 


6T 


18° 

/ 

sin   tan   cot   cos 

/ 

O 

.30  902  .32  492  3.0777  .95  106 

6O 

1 

.30  929  .32  524  3.0746  .95  097 

59 

2 

.30957  .32556  3.0716  .95088 

58 

3 

.30985  .32588  3.0686  .95079 

57 

4 

.31012  .32621  3.0655  .95070 

56 

5 

.31  040  .32  653  3.0625  .95  061 

55 

6 

.31068  .32685  3.0595  .95052 

54 

7 

.31  095  .32  717  3.0565  .95  043 

53 

8 

.31  123  .32  749  3.0535  .95  033 

52 

9 

.31151  .32782  3.0505  .95024 

51 

10 

.31  178  .32  814  3.0475  .95  015 

50 

11 

.31  206  .32  846  3.0445  .95  006 

49 

12 

.31233  .32878  3.0415  .94997 

48 

13 

.31261  .32911  3.0385  .94988 

47 

14 

.31289  .32943  3.0356  .94979 

46 

15 

.31316  .32975  3.0326  .94970 

45 

>16 

.31  344  .33  007  3.0296  .94  961 

44 

17 

.31  372  .33  040  3.0267  .94  952 

43 

18 

.31399  .33072  3.0237  .94943 

42 

19 

.31  427  .33  104  3.0208  .94  933 

41 

2O 

.31  454  .33  136  3.0178  .94924 

40 

21 

.31482  .33169  3.0149  .94915 

39 

22 

.31510  .33201  3.0120  .94906 

38 

23 

.31  537  .33  233  3.0090  .94  897 

37 

24 

.31565  .33266  3.0061  .94888 

36 

25 

.31593  .33298  3.0032  .94878 

35 

26 

.31620  .33330  3.0003  .94869 

34 

27 

.31648  .33363  2.9974  .94860 

33 

28 

.31675  .33395  2.9945  .94851 

32 

29 

r.  31  703  .33427  2.9916  .94842 

31 

30 

\31730  .33460  2.9887  .94832 

30 

31 

.31  758  .33  492  2.9858  .94  823 

29 

32 

.31  786  .33  524  2.9829  .94  814 

28 

33 

.31  813  .33  557  2.9800  .94  805 

27 

34 

.31  841  .33  589  2.9772  .94  795 

26 

35 

.31  868  .33  621  2.9743  .94  786 

25 

36 

.31  896  .33  654  2.9714  .94  777 

24 

37 

.31923  .33686  2.9686  .94768 

23 

38 

.31951  .33718  2.9657  .94758 

22 

39 

.31  979  .33  751  2.9629  .94  749 

21 

4O 

.32  006  .33  783  2.9600  .94  740 

2O 

41 

.32  034  .33  816  2.9572  .94  730 

19 

42 

.32  061  .33  848  2.9544  .94  721 

18 

43 

.32089  .33881  2.9515  .94712 

17 

44 

.32116  .33913  2.9487  .94702 

16 

45 

.32144  .33945  2.9459  .94693 

15 

46 

.32  171  .33  978  2.9431  .94  684 

14 

47 

.32  199  .34  010  2.9403  .94  674 

13 

48 

.32227  .34043  2.9375  .94665 

12 

49 

.32  254  .34  075  2.9347  .94  656 

11 

50 

.32282  .34108  2.9319  .94646 

1O 

51 

.32  309  .34  140  2.9291  .94  637 

9 

52 

.32  337  .34  173  2.9263  .94  627 

8 

53 

.32364  .34205  2.9235  .94618 

7 

54 

.32392  .34238  2.9208  .94609 

6 

55 

.32  419  .34  270  2.9180  .94  599 

5 

56 

.32447  .34303  2.9152  .94590 

4 

57 

.32474  .34335  2.9125  .94580 

3 

58 

.32502  .34368  2.9097  .94571 

2 

59 

.32  529  .34  400  2.9070  .94  561 

1 

6O 

.32557  .34433  2.9042  .94552 

0 

/ 

cos    cot   tan    sin 

/ 

71° 

19° 

/ 

sin   tan   cot   cos 

/ 

o 

.32557  .34433  2.9042  .94552 

6O 

1 

.32  584  .34  465  2.9015  .94  542 

59 

2 

.32612  .34498  2.8987  .94533 

58 

3 

.32  639  .34  530  2.8960  .94  523 

57 

4 

.32667  .34563  2.8933  .94514 

56 

5 

.32  694  .34  596  2.8905  .94  504 

55 

6 

.32  722  .34  628  2.8878  .94  495 

54 

7 

.32749  .34661  2.8851  .94485 

53 

8 

.32  777  .34  693  2.8824  .94  476 

52 

9 

.32  804  .34  726  2.8797  .94  466 

51 

10 

.32832  .34758  2.8770  .94457 

5O 

11 

.32859  .34791  2.8743  .94447 

49 

12 

.32887  .34824  2.8716  .94438 

48 

13 

.32914  .34856  2.8689  .94428 

47 

14 

.32942  .34889  2.8662  .94418 

46 

15 

.32969  .34922  2.8636  .94409 

45 

16 

.32997  .34954  2.8609  .94399 

44 

17 

.33  024  .34  987  2.8582  .94  390 

43 

18 

.33051  .35020  2.8556  .94380 

42 

19 

.33  079  .35  052  2.8529  .94  370 

41 

20 

.33  106  .35  085  2.8502  .94  361 

4O 

21 

.33134  .35118  2.8476  .94351 

39 

22 

.33  161  .35  150  2.8449  .94  342 

38 

23 

.33  189  .35  183  2.8423  .94  332 

37 

24 

.33216  .35216  2.8397  .94322 

36 

25 

.33  244  .35  248  2.8370  .94  313 

35 

26 

.33  271  .35  281  2.8344  .94  303 

34 

27 

.33298  .35314  2.8318  .94293 

33 

28 

.33  326  .35  346  2.8291  .94  284 

32 

29 

.33  353  .35  379  2.8265  .94  274 

31 

3O 

.33  381  .35  412  2.8239  .94  264 

30 

31 

.33  408  .35  445  2.8213  .94  254 

29 

32 

.33436  .35477  2.8187  .94245 

28 

33 

.33463  .35510  2.8161  .94235 

27 

34 

.33  490  .35  543  2.8135  .94  225 

26 

35 

.33  518  .35  576  2.8109  .94  215 

25 

36 

.33  545  .35  608  2.8083  .94  206 

24 

37 

.33  573  .35  641  2.8057  .94  196 

23 

38 

.33  600  .35  674  2.8032  .94  186 

22 

39 

.33  627  .35  707  2.8006  .94  176 

21 

4O 

.33  655  .35  740  2.7980  .94  167 

2O 

41 

.33  682  .35  772  2.7955  .94  157 

19 

42 

.33  710  .35  805  2.7929  .94  147 

18 

43 

.33  737  .35  838  2.7903  .94  137 

17 

44 

.33  764  .35  871  2.7878  .94  127 

16 

45 

.33  792  .35  904  2.7852  .94  118 

15 

46 

.33  819  .35  937  2.7827  .94  108 

14 

47 

.33846  .35969  2.7801  .94098 

13 

48 

.33  874  .36  002  2.7776  .94  088 

12 

49 

.33901  .36035  2.7751  .94078 

11 

50 

.33  929  .36  068  2.7725  .94  068 

1O 

51 

.33  956  .36  101  2.7700  .94  058 

9 

52 

.33  983  .36  134  2.7675  .94  049 

8 

53 

.34011  .36167  2.7650  .94039 

7 

54 

.34  038  .36  199  2.7625  .94  029 

6 

55 

.34065  .36232  2.7600  .94019 

5 

56 

.34093  .36265  2.7575  .94009 

4 

57 

.34120  .36298  2.7550  .93999 

3 

58 

.34147  .36331  2.7525  .93989 

2 

59 

.34  175  .36  364  2.7500  .93  979 

1 

6O 

.34  202  .36  397  2.7475  .93  969 

0 

/ 

cos    cot   tan    sin 

/ 

70° 

68 


NATURAL   FUNCTIONS 


20° 

/ 

sin   tan   cot   cos 

/ 

o 

.34202  .36397  2.7475  .93969 

6O 

1 

.34  229  .36  430  2.7450  .93  959 

59 

2 

.34  257  .36  463'  2.7425  .93  949 

58 

3 

.34  284  .36  496  2.7400  .93  939 

57 

4 

.34311  .36529  2.7376  .93929 

56 

5 

.34339  .36562  2.7351  .93919 

55 

6 

.34  366  .36  595  2.7326  .93  909 

54 

7 

.34  393  .36  628  2.7302  .93  899 

53 

8 

.34  421  .36  661  2.7277  .93  889 

52 

9 

.34  448  .36  694  2.7253  .93  879 

51 

10 

.34475  .36727  2.7228  .93869 

50 

11 

.34  503  .36  760  2.7204  .93  859 

49 

12 

.34  530  .36  793  2.7179  .93  849 

48 

13 

.34  557  .36  826  2.7155  .93  839 

47 

14 

.34584  .36859  2.7130  .93829 

46 

15 

.34612  .36892  2.7106  .93819 

45 

16 

.34  639  .36  925  2.7082  .93  809 

44 

17 

.34666  .36958  2.7058  .93799 

43 

18 

.34  694  .36  991  2.7034  .93  789 

42 

19 

.34  721  .37  024  2.7009  .93  779 

41 

20 

.34748  .37057  2.6985  .93769 

4O 

21 

.34.775  .37090  2.6961  .93759 

39 

22 

.34  803  .37  123  2.6937  .93  748 

38 

23 

.34830  .37157  2.6913  .93738 

37 

24 

.34  857  .37  190  2.6889  .93  728 

36 

25 

.34  884  .37  223  2.6865  .93  718 

35 

26 

.34  912  .37  256  2.6841  .93  708 

34 

27 

.34  939  .37  289  2.6818  .93  698 

33 

28 

.34  966  .37  322  2.6794  .93  688 

32 

29 

.34  993  .37  355  2.6770  .93  677 

31 

30 

.35  021  .37  388  2.6746  .93  667 

30 

31 

.35048  .37422  2.6723  .93657 

29 

32 

.35  075  '.37  455  2.6699  .93  647 

28 

33 

.35  102  .37  488  2.6675  .93  637 

27 

34 

.35  130  .37  521  2.6652  .93  626 

26 

35 

.35157  .37554  2.6628  .93616 

25 

36 

.35  184  .37  588  2.6605  .93  606 

24 

37 

.35211  .37621  2.6581  .93596 

23 

38 

.35  239  .37  654  2.6558  .93  585 

22 

39 

.35  266  .37  687  2.6534  .93  575 

21 

40 

.35  293  .37  720  2.6511  .93  565 

20 

41 

.35  320  .37  754  2.6488  .93  555 

19 

42 

;35  347  .37  787  2.6464  .93  544 

18 

43 

.35  375  .37  820  2.6441  .93  534 

17 

44 

.35402  .37853  2.6418  .93524 

16 

45 

.35429  .37887  2.6395  .93514 

15 

46 

.35  456  .37  920  2.6371  .93  503 

14 

47 

.35  484  .37  953  2.6348  .93  493 

13 

48 

.35  511  .37  986  2.6325  .93  483 

12 

49 

.35  538  .38  020  2.6302  .93  472 

11 

50 

.35565  .38053  2.6279  ;93  462 

1O 

51 

.35  592  .38  086  2.6256  .93  452 

9 

52 

.35  619  .38  120  2.6233  .93  441 

8 

53 

35647  .38153  2.6210  .93431 

7 

54 

.35  674  .38  186  2.6187  .93  420 

6 

55 

.35  701  .38  220  2.6165  .93  410 

5 

56 

.35  728  .38  253  2.6142  .93  400 

4 

57 

.35  755  .38  286  2.6119  .93  389 

3 

58 

.35  782  .38320  2.6096  .93379 

2 

59 

.35  810  .38  353  2.6074  .93  368 

1 

6O 

.35837  .38386  2.6051  .93358 

O 

/ 

cos    cot   tan    sin 

/ 

69° 

21° 

/ 

sin   tan   cot   cos 

/ 

O 

.35837  .38386  2.6051  .93358 

6O 

1 

.35  864  .38  420  2.6028  .93  348 

59 

2 

.35891  .38453  2.6006  .93337 

58 

3 

.35  918  .38  487  2.5983  .93  327 

57 

4 

.35  945  .38  520  2.5961  .93  316 

56 

5 

.35  973  .38  553  2.5938  .93  306 

55 

6 

.36  000  .38  587  2.5916  .93  295 

54 

7 

.36  027  .38  620  2.5893  .93  285 

53 

8 

.36054  .38654  2.5871  .93274 

52 

9 

.36  081  .38  687  2.5848  .93  264 

51 

1O 

.36  108  .38  721  2.5826  .93  253 

5O 

11 

.36  135  .38  754  2.5804  .93  243 

49 

12 

.36  162  .38  787  2.5782  .93  232 

48 

13 

.36  190  .38  821  2.5759  .93  222 

47 

14 

.36217  .38854  2.5737  .93211 

46 

15 

.36  244  .38  888  2.5715  .93  201 

45 

16 

.36  271  .38  921  2.5693  .93  190 

44 

17 

.36  298  .38  955  2.5671  .93  180 

43 

18 

.36  325  .38  988  2.5649  .93  169 

42 

19 

.36  352  .39  022  2.5627  .93  159 

41 

20 

.36  379  .39  055  2.5605  .93  148 

4O 

21 

.36406  .39089  2.5583  .93137 

39 

22 

.36  434  .39  122  2.5561  .93  127 

38 

23 

.36461  .39156  2.5539  .93  116 

37 

24 

.36488  .39190  2.5517  .93106 

36 

25 

.36515  .39223  2.5495  .93095 

35 

26 

.36542  .39257  2.5473  .93084 

34 

27 

.36  569  .39  290  2.5452  .93  074 

33 

28 

.36  596  .39  324  2.5430  .93  063 

32 

29 

.36623  .39357  2.5408  .93052 

31 

30 

.36  650  .39  391  2.5386  .93  042 

30 

31 

.36677  .39425  2.5365  .93031 

29 

32 

.36  704  .39  458  2.5343  .93  020 

28 

33 

.36731  .39492  2.5322  .93010 

27 

34 

.36  758  .39  526  2.5300  .92  999 

26 

35 

.36785  .39559  2.5279  .92988 

25 

36 

.36812  .39593  2.5257  .92978 

24 

37 

.36  839  .39  626  2.5236  .92  967 

23 

38 

.36867  .39660  2.5214  .92956 

22 

39 

.36894  .39694  2.5193  .92945 

21 

4O 

.36921  .39727  2.5172  .92935 

2O 

41 

.36948  .39761  2.5150  .92924 

19 

42 

.36975  .39795  2.5129  .92913 

18 

43 

.37002  .39829  2.5108  .92902 

17 

44 

.37029  .39862  2.5086  .92892 

16 

45 

.37  056  .39  896  2.5065  .92  881 

15 

46 

.37  083  .39  930  2.5044  .92  870 

14 

47 

.37110  .39963  2.50Z3  .92859 

13 

48 

.37  137  .39  997  2.5002  .92  849 

12 

49 

.37  164  .40  031  2.4981  .92  838 

11 

5O 

.37191  .40065  2.4960  .92827 

10 

51 

.37218  .40098  2.4939  .92816 

9 

52 

.37  245  .40  132  2.4918  .92  805 

8 

53 

.37  272  .40  166  2.4897  .92  794 

7 

54 

.37  299  .40  200  2.4876  .92  784 

6 

55 

.37  326  .40  234  2.4855  .92  773 

5 

56 

.37  353  .40  267  2.4834  .92  762 

4 

57 

.37380  .40301  2.4813  .92751 

3 

58 

.37407  .40335  2.4792  .92740 

2 

59 

.37  434  .40  369  2.4772  .92  729 

1 

6O 

.37461  .40403  2.4751  .92718 

0 

/ 

cos    cot   tan    sin 

/ 

68° 

NATURAL  FUNCTIONS 


69 


22° 

/ 

sin        tan        cot        cos 

/ 

o 

.37461  .40403  2.4751  .92718 

60 

1 

.37  488  .40  436  2.4730  .92  707 

59 

2 

.37515  .40470  2.4709  .92697 

58 

3 

.37  542  .40  504  2.4689  .92  686 

57 

4 

.37  569  .40  538  2.4668  .92  675 

56 

5 

.37595  .40572  2.4648  .92664 

55 

6 

.37622  .40606  2.4627  .92653 

54 

7 

.37649  .40640  2.4606  .92642 

53 

8 

.37676  .40674  2.4586  .92631 

52 

9 

.37  703  .40  707  2.4566  .92  620 

51 

1C 

.37  730  .40  741  2.4545  .92  609 

50 

11 

.37757  .40775  2.4525  .92598 

49 

12 

.37  784  .40  809  2.4504  .92  587 

48 

13 

.37811  .40843  2.4484  .92576 

47 

14 

.37  838  .40  877  2.4464  .92  565 

46 

15 

.37865  .40911  2.4443  .92554 

45 

16 

.37892  .40945  2.4423  .92543 

44 

17 

.37  919  .40  979  2.4403  .92  532 

43 

18 

.37  946  .41  013  2.4383  .92  521 

42 

19 

.37973  .41047  2.4362  .92510 

41 

20 

.37  999  .41  081  2.4342  .92  499 

4O 

21 

.38  026  .41  115  2.4322  .92  488 

39 

22 

.38  053  .41  149  2.4302  .92  477 

38 

23 

.38  080  .41  183  2.4282  .92  466 

37 

24 

.38  107  .41  217  2.4262  .92  455 

36 

25 

.38134  .41251  2.4242  .92444 

35 

26 

.38161  .41285  2.4222  .92432 

34 

27 

.38  188  .41  319  2.4202  .92  421 

33 

28 

.38215  .41353  2.4182  .92410 

32 

29 

.38  241  .41  387  2.4162  .92  399 

31 

30 

.38268  .41421  2.4142  .92388 

30 

31 

.38295  .41455  2.4122  .92377 

29 

32 

.38  322  .41  490  2.4102  .92  366 

28 

33 

.38349  .41524  2.4083  .92355 

27 

34 

.38  376  .41  558  2.4063  .92  343 

26 

35 

.38  403  .41  592  2.4043  .92  332 

25 

36 

.38430  .41626  2.4023  .92321 

24 

37 

.38  456  .41  660  2.4004  .92  310 

23 

38 

.38  483  .41  694  2.3984  .92  299 

22 

39 

.38  510  .41  728  2.3964  .92  287 

21 

40 

.38  537  .41  763  2.3945  .92  276 

2O 

41 

.38  564  .41  797  2.3925  .92  265 

19 

42 

.38  591  .41  831  2.3906  .92  254 

18 

43 

.38  617  .41  865  2.3886  .92  243 

17 

44 

.38644  .41899  2.3867  .92231 

16 

45 

.38671  .41933  2.3847  .92220 

15 

46 

.38  698  .41  968  2.3828  .92  209 

14 

47 

.38  725  .42  002  2.3808  .92  198 

13 

48 

.38  752  .42  036  2.3789  .92  186 

12 

49 

.38  778  .42  070  2.3770  .92  175 

11 

5O 

.38  805  .42  105  2.3750  .92  164 

1O 

51 

.38  832  .42  139  2.3731  .92  152 

9 

52 

.38  859  .42  173  2.3712  .92  141 

8 

53 

.38  886  .42  207  2.3693  .92  130 

7 

54 

.38912  .42242  2.3673  .92119 

6 

55 

.38  939  .42  276  2.3654  .92  107 

5 

56 

.38966  .42310  2.3635  .92096 

4 

57 

.38993  .42345  2.3616  .92085 

3 

58 

.39020  .42379  2.3597  .92073 

2 

59 

.39046  .42413  2.3578  .92062 

1 

6O 

.39073  .42447  2.3559  .92050 

O 

/ 

cos        cot        tan        sin 

/ 

67° 

23° 

/ 

sin   tan   cot   cos 

/ 

0 

.39073  .42447  2.3559  .92050 

60 

1 

.39  100  .42  482  2.3539  .92  039 

59 

2 

.39  127  .42  516  2.3520  .92  028 

58 

3 

.39153  .42551  2.3501  .92016 

57 

4 

.39  180  .42  585  2.3483  .92  005 

56 

5 

.39207  .42619  2.3464  .91994 

55 

6 

.39  234  .42  654  2.3445  .91  982 

54 

7 

.39  260  .42  688  2.3426  .91  971 

53 

8 

.39  287  .42  722  2.3407  .91  959 

52 

9 

.39  314  .42  757  2.3388  .91  948 

51 

10 

.39  341  .42  791  2.3369  .91  936 

5O 

11 

.39  367  .42  826  2.3351  .91  925 

49 

12 

.39  394  .42  860  2.3332  .91  914 

48 

13 

.39421  .42894  2.3313  .91902 

47 

14 

.39  448  .42  929  2.3294  .91  891 

46 

15 

.39474  .42963  2.3276  .91879 

45 

16 

.39  501  .42  998  2.3257  .91  868 

44 

17 

.39  528  .43  032  2.3238  .91  856 

43 

18 

.39  555  .43  067  2.3220  .91  845 

42 

19 

.39  581  .43  101  2.3201  .91  833 

41 

2O 

.39  608  .43  136  2.3183  .91  822 

4O 

21 

.39635  .43170  2.3164  .91*810 

39 

•22 

.39  661  .43  205  2.3146  .91  799 

38 

23 

.39  688  .43  239  2.3127  .91  787 

37 

24 

.39  715  .43  274  2.3109  .91  775 

36 

25 

.39  741  .43  308  2.3090  .91  764 

35 

26 

.39  768  .43  343  2.3072  .91  752 

34 

27 

.39  795  .43  378  2.3053  .91  741 

33 

28 

.39  822  .43  412  2.3035  .91  729 

32 

29 

.39  848  .43  447  2.3017  .91  718 

31 

30 

.39  875  .43  481  2.2998  .91  706 

30 

31 

.39902  .43516  2.2980  .91694 

29 

32 

.39  928  .43  550  2.2962  .91  683 

28 

33 

.39  955  .43  585  2.2944  .91  671 

27 

34 

.39  982  .43  620  2.2925  .91  660 

26 

35 

.40  008  .43  654  2.2907  .91  648 

25 

36 

.40  035  .43  689  2.2889  .91  636 

24 

37 

.40  062  .43  724  2.2871  .91  625 

23 

38 

.40  088  .43  758  2.2853  .91  613 

22 

39 

.40  115  .43  793  2.2835  .91  601 

21 

4O 

.40  141  .43  828  2.2817  .91  590 

2O 

41 

.40  168  .43  862  2.2799  .91  578 

19 

42 

.40  195  .43  897  2.2781  .91  566 

18 

43 

.40  221  .43  932  2.2763  .91  555 

17 

44 

.40  2H8  .43  966  2.2745  .91  543 

16 

45 

.40  275  .44  001  2.2727  .91  531 

15 

46 

.40  301  .44  036  2.2709  .91  519 

14 

47 

.40  328  .44  071  2.2691  .91  508 

13 

48 

.40355  .44"  105  2.2673  .91496 

12 

49 

.40381  .44140  2.2655  .91484 

11 

50 

.40408  .44175  2.2637  .91472 

1O 

51 

.40434  .44210  2.2620  .91461 

9 

52 

.40461  .44244  2.2602  .91449 

8 

53 

.40  488  .44  279  2.2584  .91  437 

7 

54 

.40514  .44314  2.2566  .91425 

6 

55 

.40  541  .44  349  2.2549  .91  414 

5 

56 

.40567  .44384  2.2531  .91402 

4 

57 

.40594  .44418  2.2513  .91390 

3 

58 

.40621  .44453  2.2496  .91378 

2 

59 

.40  647  .44  488  2.2478  .91  366 

1 

60 

.406.74  .44  523  2.2460  .91  355 

0 

/ 

cos    cot   tan    sin 

/ 

66° 

70 


NATURAL   FUNCTIONS 


24° 

/ 

sin    tan    cot   cos 

/ 

o 

.40674  .44523  2.2460  .91355 

6O 

1 

.40  700  .44  558  2.2443  .91  343 

59 

2 

.40727  .44593  2.2425  .91331 

58 

3 

.40753  .44627  2.2408  .91319 

57 

4 

.40  780  .44  662  2.2390  .91  307 

56 

5 

.40  806  .44  697  2.2373  .91  295 

55 

6 

.40  833  .44  732  2.2355  .91  283 

54 

7 

.40  860  .44  767  2.2338  .91  272 

53 

8 

.40  886  .44  802  2.2320  .91  260 

52 

9 

.40913  .44837  2.2303  .91248 

51 

1C 

.40  939  .44  872  2.2286  .91  236 

5O 

11 

.40  966  .44  907  2.2268  .91  224 

49 

12 

.40992  .44942  2.2251-  .91212 

48 

13 

.41019  .44977  2.2234  .91200 

47 

14 

.41  045  .45  012  2.2216  .91  1S8 

46 

15 

.41  072  .45  047  2.2199  .91  176 

45 

16 

.41  098  .45  082  2.2182  .91  164 

44 

17 

.41  125  .45  117  2.2165  .91  152 

43 

18 

.41  151  .45  152  2.2148  .91  140 

42 

19 

.41  178  .45  187  2.2130  .91  128 

41 

2O 

.41204  .45222  2.2113  .91116 

40 

21 

.41  231  .45  257  2.2096  .91  104 

39 

22 

.41  257  .45  292  2.2079  .91  092 

38 

23 

.41  284  .45  327  2.2062  .91  080 

37 

24 

.41  310  .45  362  2.2045  .91  068 

36 

25 

.41  337  .45  397  2.2028  .91  056 

35 

26 

.41  363  .45  432  2.2011  .91  044 

34 

27 

.41  390  .45  467  2.1994  .91  032 

33 

28 

.41  416  .45  502  2.1977  .91  020 

32 

29 

.41  443  .45  538  2.1960  .91  008 

31 

30 

.41469  .45573  2.1943  .90996 

30 

31 

.41  496  .45  608  2.1926  .90984 

29 

32 

.41  522  .45  643  2.1909  .90  972 

28 

33 

.41549  .45678  2.1892  .90960 

27 

34 

.41575  .45  713  2.1876  .90948 

26 

35 

.41602  .45748  2.1859  .90936 

25 

36 

.41  628  .45  784  2.1842  .90  924 

24 

37 

.41655  .45819  2.1825  .90911 

23 

38 

.41  681  .45  854  2.1808  .90  899 

22 

39 

.41  707  .45  889  2.1792  .90  887 

21 

40 

.41  734  .45  924  2.1775  .90  875 

20 

41 

.41  760  .45  960  2.1758  .90  863 

19 

42 

.41787  .45995  2.1742  .90851 

18 

43 

.41813  .46030  2.1725  .90839 

17 

44 

.41  840  .46  065  2.1708  .90  826 

16 

45 

.41  866  .46  101  2.1692  .90  814 

15 

46 

.41  892  .46  136  2.1675  .90  802 

14 

47 

.41  919  .46  171  2.1659  .90  790 

13 

48 

.41  945  .46  206  2.1642  .90  778 

12 

49 

.41  972  .46  242  2.1625  .90  766 

11 

5O 

.41998  .46277  2.1609  .90753 

10 

51 

.42  024  .46  312  2.1592  .90  741 

9 

52 

.42051  .46348  2.1576  .90729 

8 

53 

.42077  .46383  2.1560  .90717 

7 

54 

.42104  .46418  2.1543  .90704 

6 

55 

.42  130  .46  454  2.1527  .90  692 

5 

56 

.42156  .46489  2.1510  .90680 

4 

57 

.42  183  .46  525  2.1494  .90  668 

3 

58 

.42  209  .46  560  2.1478  .90  655 

2 

59 

.42235  .46595  2.1461  .90643 

1 

60 

.42262  .46631  2.1445  .90631 

0 

/ 

cos    cot    tan    sin 

/ 

65° 

25° 

/ 

sin    tan    cot    cos 

/ 

0 

.42262  .46631  2.1445  .90631 

60 

1 

.42  288  .46  666  2.1429  .90  618 

59 

2 

.42315  .46702  2.1413  .90606 

58 

3 

.42341  .46737  2.1396  .90594^ 

57 

4 

.42367  .46772  2.1380  .90582 

56 

5 

.42  394  .46  808  2.1364  .90  569 

55 

6 

.42420  .46843  2.1348  .90557 

54 

7 

.42  446  .46  879  2.1332  .90  545 

53 

*8 

.42473  .46914  2.1315  .90532 

52 

%9 

.42499  .46950  2.1299  .90520 

51 

10 

.42525  .46985  2.1283  .90507 

50 

11 

.42552  .47021  2.1267  .90495 

49 

12 

.42578  .47056  2.1251  .90483 

48 

13 

.42604  .47092  2.1235  .90470 

47 

14 

.42631  .47128  2.1219  .90458 

46 

15 

.42657  .47163  2.1203  .90446 

45 

16 

.42683  .47199  2.1187  .90433 

44 

17 

.42709  .47234  2.1171  .90421 

43 

18 

.42736  .47270  2.1155  .90408 

42 

19 

.42762  .47305  2.1139  .90396 

41 

2O 

.42788  .47341  2.1123  .90383 

40 

21 

.42815  .47377  2.1107  .90371 

39 

22 

.42841  .47412  2.1092  .90358 

38 

23 

.42867  .47448  2.1076  .90346 

37 

24 

.42  894  .47  483  2.1060  .90  334 

36 

25 

.42920  .47519  2.1044  .90321 

35 

26 

.42946  .47555  2.1028  .90309 

34 

27 

.42  972  .47  590  2.1013  .90  296 

33 

28 

.42999  .47626  2.0997  .90284 

32 

29 

.43025  .47662  2.0981  .90271 

31 

3O 

.43051  .47698  2.0965  .90259 

3O 

31 

.43  077  .47  733  2.0950  .90  246 

29 

32 

.43  104  .47  769  2.0934  .90  233 

28 

33 

.43  130  .47  805  2.091S  .90  221 

27 

34 

.43  156  .47  840  2.0903  .90  208 

26 

35 

.43  182  .47  876  2.0887  .90  196 

25 

36 

.43  209  .47  912  2.0872  .90  183 

24 

37 

.43235  .47948  2.0856  .90171 

23 

38 

.43261  .47984  2.0840  .90158 

22 

39 

.43  287  .48  019  2.0825  .90  146 

21 

40 

.43313  .48055  2.0809  .90133 

2O 

41 

.43  340  .48  091  2.0794  .90  120 

19 

42 

.43  366  .48  127  2.0778  .90  108 

18 

43 

.43  392  .48  163  2.0763  .90  095 

17 

44 

.43  418  .48  198  2.0748  .90  082 

16 

45 

.43445  .48234  2.0732  .90070 

15 

46 

.43471  .48270  2.0717  .90057 

14 

47 

.43497  .48306  2.0701  .90045 

13 

48 

.43  523  .48  342  2.0686  .90  032 

12 

49 

.43549  .48378  2.0671  .90019 

11 

50 

.43575  .48414  2.0655  .90007 

1O 

51 

.43  602  .48  450  2.0640  .89  994 

9 

52 

.43628  .48486  2.0625  .89981 

8 

53 

.43  654  .48  521  2.0609  .89  968 

7 

54 

.43680  .48557  2.0594  .89956 

6 

55 

.43  706  .48  593  2.0579  .89  943 

5 

56 

.43  733  .48  629  2.0564  .89  930 

4 

57 

.43  759  .48  665  2.0549  .89  918 

3 

58 

.43  785  .48  701  2.0533  .89  905 

2 

59 

.43  811  .48  737  2.0518  .89  892 

1 

6O 

.43  837  .48  773  2.0503  .89  879 

0 

/ 

cos   cot   tan   sin 

/ 

64° 

NATURAL   FUNCTIONS 


71 


26° 

/ 

sin        tan        cot        cos 

/ 

0 

.43837  .48773  2.0503  .89879 

60 

1 

.43863  .48809  2.0488  .89867 

59 

2 

.43  889  .48  845  2.0473  .89  854 

58 

3 

.43916  .48881  2.0458  .89841 

57 

4 

.43942  .48917  2.0443  .89828 

56 

5 

.43968  .48953  2.0428  .89816 

55 

6 

.43  994  .48  989  2.0413  .89  803 

54 

7 

.44020  .49026  2.0398  .89790 

53 

8 

.44  046  .49  062  2.0383  .89  777 

52 

9 

.44  072  .49  098  2.0368  .89  764 

51 

1O 

.44098  .49134  2.0353  .89752 

5O 

11 

.44  124  .49  170  2.0338  .89  739 

49 

12 

.44151  .49206  2.0323  .89726 

48 

13 

.44177  .49242  2.0308  .89713 

47 

14 

.44203  .49278  2.0293  .89700 

46 

15 

.44229  .49315  2.0278  .89687 

45 

16 

.44255  .49351   2.0263  .89674 

44 

17 

.44281  .49387  2.0248  .89662 

43 

18 

.44307  .49423  2.0233  .89649 

42 

19 

.44333  .49459  2.0219  .89636 

41 

2O 

.44359  .49495  2.0204  .89623 

4O 

21 

.44385  .49532  2.0189  .89610 

39 

22 

.44411  .49568  2.0174  .89597 

38 

23 

.44437  .49604  2.0160  .89584 

37 

24 

.44  464  .49  640  2.0145  .89  571 

36 

25 

.44  490  .49  677  2.0130  .89  558 

35 

26 

.44  516  .49  713  2.0115  .89  545 

34 

27 

.44  542  .49  749  2.0101  .89  532 

33 

28 

.44568  .49786  2.0086  .89519 

32 

29 

.44594  .49822  2.0072  .89506 

31 

'3O 

.44620  .49858  2.0057  .89493 

30 

31 

.44646  .49894  2.0042  .89480 

29 

32 

.44672  .49931  2.0028  .89467 

28 

33 

.44698  .49967  2.0013  .89454 

27 

34 

.44  724  .50  004  1.9999  .89  441 

26 

35 

.44750  .50040  1.9981-  .89428 

25 

36 

.44  776  .50  076  1.9970  .89  415 

24 

37 

.44802  .50113  1.9955  .89402 

23 

38 

.44  828  .50  149  1.9941  .89  389 

22 

39 

.44  854  .50  185  1.9926  .89  376 

21 

40 

.44880  .50222  1.9912  .89363 

20 

41 

.44906  .50258  1.9897  .89350 

19 

42 

.44932  .50295  1.9883  .89337 

18 

43 

.44958  .50331  1.9868  .89324 

17 

44 

.44984  .50368  1.9854  .89311 

16 

45 

.45010  .50404  1.9840  .89298 

15 

46 

.45036  .50441  1.9825  .89285 

14 

47 

.45062  .50477  1.9811  .89272 

13 

48 

.45  088  .50  514  1.9797  .89  259 

12 

49 

.45114  .50550  1.9782  .89245 

11 

5O 

.45  140  .50  587  1.9768  .89  232 

10 

51 

.45  166  .50  623  1.9754  .89  219 

9 

52 

.45192  .50660  1.9740  .89206 

8 

53 

.45218  .50696  1.9725  .89193 

7 

54 

.45243  .50733  1.9711  .89180 

6 

55 

.45269  .50769  1.9697  .89167 

5 

56 

.45  295  .50  806  1.9683  .89  153 

4 

57 

.45321  .50843  1.9669  .89140 

3 

58 

.45347  .50879  1.9654  .89127 

2 

59 

.45  373  .50  916  1.9640  .89  114 

1 

60 

.45399  .50953  1.9626  .89101 

O 

/ 

cos        cot        tan        sin 

/ 

63° 

27° 

/ 

sin    tan    cot    cos 

f 

0 

.45  399  .50  953  1.9626  .89  101 

60 

1 

.45425  .50989  1.9612  .89087 

59 

2 

.45451  .51026  1.9598  .89074 

58 

3 

.45477  .51063  1.9584  .89061 

57 

4 

.45503  .51099  1.9570  .89048 

56 

5 

.45529  .51136  1.9556  .89035 

55 

6 

.45554  .51173  1.9542  .89021 

54 

'7 

.45580  .51209  1.9528  .89008 

53 

8 

.45606  .51246  1.9514  .88995 

52 

9 

.45  632  .51  283  1.9500  .88  981 

51 

1O 

.45658  .51319  1.9486  .88968 

5O 

11 

.45  684  .51  356  1.9472  .88  955 

49 

12 

.45  7JO  .51393  1.9458  .88942 

48 

13 

.45736  .51430  1.9444  .88928 

47 

14 

.45762  .51467  1.9430  .88915 

46 

15 

.45  787  .51  503  1.9416  .88902 

45 

16 

.45813  .51  540  1.9402  .88888 

44 

•  17 

.45839  .51577  1.9388  .88875 

43 

18 

.45865  .51  614  1.9375  .88862 

42 

19 

.45891  .51651  1.9361  .88848 

41 

20 

.45917  .51688  1.9347  .88835 

4O 

21 

.45942  .51  724  1.9333  .88822 

39 

22 

.45968  .51  761  1.9319  .88808 

38 

23 

.45  994  .51  798  1.9306  .88  795 

37 

24 

.46020  .51  835  1.9292  .88782 

36 

25 

.46046  .51872  1.9278  .88768 

35 

26 

.46072  .51909  1.9265  .88755 

34 

27 

.46097  .51946  1.9251  .88741 

33 

28 

.46123  .51983  1.9237  .88728 

32 

29 

.46149  .52020  1.9223  .88715 

31 

30 

.46  175  .52  057  1.9210  .88  701 

3O 

31 

.46201  .52094  1.9196  .88688 

29 

32 

.46226  .52131  1.9183  .88674 

28 

33 

.46252  .52168  1.9169  .88661 

27 

34 

.46278  .52205  1.9155  .88647 

26 

35 

.46304  .52242  1.9142  .88634 

25 

36 

.46330  .52279  1.9128  .88620 

24 

37 

.46355  .52316  1.9115  .88607 

23 

38 

.46381  .52353  1.9101  .88593 

22 

39 

.46407  .52390  1.9088  .88580 

21 

40 

.46433  .52427  1.9074  .88566 

2O 

41 

.46458  .52464  1.9061  .88553 

19 

42 

.46484  .52501  1.9047  .88539 

18 

43 

.46  510  .52  538  1.9034  .88  526 

17 

44 

.46536  .52575  1.9020  .88512 

16 

45 

.46  561  .52  613  1.9007  .88  499 

15 

46 

.46  587  .52  650  1.8993  .88  485 

14 

47 

.46  613  .52  687  1.89SO  .88  472 

13 

48 

.46  639  .52  724  1.8967  .88  458 

12 

49 

.46664  .52761  1.8953  .88445 

11 

50 

.46690  .52798  1.8940  .88431 

1O 

51 

.46716  .52836  1.8927  .88417 

9 

52 

.46742  .52873  1.8913  .88404 

8 

53 

.46767  .52910  1.8900  .88390 

71 

54 

.46793  .52947  1.8887  .88377 

6 

55 

.46819  .52985  1.8873  .88363 

5 

56 

.46  844  .53  022  1.8860  .88  349 

4 

57 

.46  870  .53  059  1.8847  .88  336 

3 

58 

.46  896  .53  096  1.8834  .88  322 

2 

59 

.46921  .53  134  1.8820  .88308 

1 

60 

.46947  .53171  1.SS07  .88  295 

0 

/ 

cos    cot   tan    sin 

/ 

62° 

72 


NATURAL   FUNCTIONS 


28° 

/ 

sin    tan    cot    cos 

/ 

o 

.46947  .53171  1.8S07  .88295 

60 

1 

.46973  .53208  1.8794  .88281 

59 

2 

.46999  .53246  1.8781  .88267 

58 

3 

.47  024  .53  283  1.8768  .88  254 

57 

4 

.47050  .53320  1.8755  .88240 

56 

5 

.47  076  .53  358  1.8741  .88  226 

55 

6 

.47  101  .53  395  1.8728  .88  213 

54 

7 

.47  127  .53  432  1.8715  .88  199 

53 

8 

.47  153  .53470  1.8702  .88  185 

52 

9 

.47  178  .53  507  1.8689  .88  172 

51 

10 

.47  204  .53  545  1.8676  .88  158 

5O 

11 

.47  229  .53  582  1.8663  .88  144 

49 

12 

.47  255  .53  620  1.8650  .88  130 

48 

13 

.47  281  .53  657  1.8637  .88  117 

47 

14 

.47  306  .53  694  1.8624  .88  103 

46 

15 

.47332  .53732  1.8611  .88089 

45 

16 

.47  358  .53  769  1.8598  .88  075 

44 

17 

.47  383  .53  807  1.8585  .88  062 

43 

18 

.47409  .53844  1.8572  .88048 

42 

19 

.47434  .53882  1.8559  .88034 

41 

2O 

.47  460  .53  920  1.8546  .88  020 

40 

21 

.47486  .53957  1.8533  .88006 

39 

22 

.47511  .53995  1.8520  .87993 

38 

23 

.47  537  .54  032  1.8507  .87  979 

37 

24 

.47562  .54070  1.8495  .87965 

36 

25 

.47588  .54307  1.8482  .87951 

35 

26 

.47  614  .54  145  1.8469  .87  937 

34 

27 

.47  639  .54  183  1.8456  .87  923 

33 

28 

.47665  .54220  18443  .87909 

32 

29 

.47  690  .54  258  1.8430  .87  896 

31 

3O 

.47  716  .54  296  1.8418  .87  882 

3O 

•  31 

.47741  .54333  1.8405  .87868 

29 

32 

.47  767  .54  371  1.8392  .87  854 

28 

33 

.47  793  .54  409  1.8379  .87  840 

27 

34 

.47  818  .54  446  1.8367  .87  826 

26 

35 

.47844  .54484  1.8354  .87812 

25 

36 

.47  869  .54  522  1.8341  .87  798 

24 

37 

.47  895  .54  560  1.8329  .87  784 

23 

38 

.47  920  .54  597  1.8316  .87  770 

22 

39 

.47  946  .54  635  1.8303  .87  756 

21 

4O 

.47971  .54673  1.8291  .87743 

2O 

41 

.47  997  .54  711  1.8278  .87  729 

19 

42 

.48022  .54748  1.8265  .87715 

18 

43 

.48048  .54786  1.8253  .87701 

17 

44 

.48073  .54824  1.8240  .87687 

16 

45 

.48099  .54862  1.8228  .87673 

15 

46 

.48124  .54900  1.8215  .87659 

14 

47 

.48150  .54938  1.8202  .87645 

13 

48 

.48175  .54975  1.8190  .87631 

12 

49 

.48201  .55013  1.8177  .87617 

11 

50 

.48226  .55051  1.8165  .87603 

1O 

51 

.48  252  .55  089  1.8152  .87  589 

9 

52 

.48  277  .55  127  1.8140  .87  575 

8 

53 

.48  303  .55  165  1.8127  .87  561 

7 

54 

.48328  .55203  1.8115  .87546 

6 

55 

.48  354  .55  241  1.8103  .87  532 

5 

56 

.48379  .55279  1.8090  .87518 

4 

57 

.48405  .55317  1.8078  .87504 

3 

58 

.48430  .55355  1.8065  .87490 

2 

59 

.48  456  .55  393  1.8053  .87  476 

1 

GO 

.48481  .55431  1.8040  .87462 

0 

/ 

cos    cot    tan    sin 

/ 

61° 

29° 

/ 

sin    tan    cot    cos 

/ 

0 

.48481  .55431  1.8040  .87462 

6O 

1 

.48506  .55469  1.8028  .87448 

59 

2 

.48  532  .55  507  1.8016  .87  434 

58 

3 

.48  557  .55  545  1.8003  .87  420 

57 

4 

.48  583  .55  583  1.7991  .87  406 

56 

5 

.48608  .55621  1.7979  .87391 

55 

6 

.48634  .55659  1.7966  .87377 

54 

7 

.48  659  .55  697  1.7954  .87  363 

53 

8 

.48684  .55  736  1.7942  .87349 

52 

9 

.48710  .55774  1.7930  .87335 

51 

1O 

.48735  .55812  1.7917  .87321 

50 

11 

.48  761  .55  850  1.7905  .87  306 

49 

12 

.48786  .55888  1.7893  .87292 

48 

13 

.48811  .55926  1.7881  .87278 

47 

14 

.48837  .55964  1.7868  .87264 

46 

15 

.48862  .56003  1.7856  .87250 

45 

16 

.48888  .56041  1.7844  .87235 

44 

17 

.48913  .56079  .7832  .87221 

43 

18 

.48938  .56117  .7820  .87207 

42 

19 

.48  964  .56  156  .7808  .87  193 

41 

2O 

.48  989  .56  194  .7796  .87  178 

4O 

21 

.49  014  .56  232  .7783  .87  164 

"  39 

22 

.49  040  .56  270  .7771  .87  150 

38 

23 

.49065  .56309  .7759  .87136 

37 

24 

.49  090  .56  347  .7747  .87  121 

36 

25 

.49  116  .56  385  .7735  .87  107 

35 

26 

.49141  .56424  .7723  .87093 

34 

27 

.49166  .56462  .7711  .87079 

33 

28 

.49192  .56  501  .7699  .87  064 

32. 

29 

.49  217  .56  539  .7687  .87  05.0 

31 

30 

.49242  .56577  .7675  .87036 

3O 

31 

.49268  .56616  .7663  .87021 

29 

32 

.49293  .56654  .7651  .87007 

28 

33 

.49318  .56693  .7639  .86993 

27 

34 

.49344  .56731  .7627  .86978 

26 

35 

.49369  .56769  1.7615  .86964 

25 

36 

.49394  .56808  1.7603  .86949 

24 

37 

.49419  .56846  1.7591  .86935 

23 

38 

.49445  .56885  1.7579  .86921 

22 

39 

.49470  .56923  1.7567  .86906 

21 

40 

.49495  .56962  1.7556  .86892 

2O 

41 

.49521  .57000  1.7544  .86878 

19 

42 

.49546  .57039  1.7532  .86863 

18 

43 

.49  571  .57  078  1.7520  .86  849 

17 

44 

.49596  .57116  1.7508  .86834 

16 

45 

.49  622  .57  155  .7496  .86  820 

15 

46 

.49647  .57  193  .7485  .86805 

14 

47 

.49  672  .57  232  .7473  .86  791 

13 

48 

.49  697  .57  271  .7461  .86  777 

12 

49 

.49  723  .57  309  .7449  .86  762 

11 

50 

.49  748  .57  348  .7437  .86  748 

1O 

51 

.49  773  .57  386  .7426  .86  733 

9 

52 

.49  798  .57  425  1.7414  .86  719 

8 

53 

.49824  .57464  1.7402  .86704 

7 

54 

.49849  .57503  1.7391  .86690 

6 

55 

.49874  .57541  1.7379  .86675 

5 

56 

.49899  .57580  1.7367  .86661 

4 

57 

.49  924  .57  619  1.7355  .86  646 

3 

58 

.49950  .57657  1.7344  .86632 

2 

59 

.49975  .57696  1.7332  .86617 

1 

60 

.50000  .57735  1.7321  .86603 

O 

/ 

cos    cot    tan    sin 

/ 

60° 

NATURAL   FUNCTIONS 


73 


30° 

/ 

sin    tan    cot    cos 

/ 

o 

.50000  .57735  1.7321  .86603 

6O 

1 

.50025  .57774  1.7309  .86588 

59 

2 

.50050  .57813  1.7297  .86573 

58 

3 

.50076  .57851  1.7286  .86559 

57 

4 

.50  101  .57  890  1.7274  .86  544 

56 

5 

.50  126  .57  929  1.7262  .86  530 

55 

6 

.50151  .57968  1.7251  .86515 

54 

7 

.50176  .58007  1.7239  .86501 

53 

8 

.50201  .58046  1.7228  .86486 

52 

9 

.50227  .58085  1.7216  .86471 

51 

10 

.50252  .58124  1.7205  .86457 

50 

11 

.50277  .58162  1.7193  .86442 

49 

12 

.50302  .58201  1.7182  .86427 

48 

13 

.50327  .58240  1.7170  .86413 

47 

14 

.50352  .58279  1.7159  .86398 

46 

15 

.50377  .58318  1.7147  .86384 

45 

16 

.50403  .58357  1.7136  .86369 

44 

17 

.50428  .58396  1.7124  .86354 

43 

18 

.50453  .58435  1.7113  .86340 

42 

!  19 

.50478  .58474  1.7102  .86325 

41 

2O 

.50503  .58513  1.7090  .86310 

40 

21 

.50  528  .58  552  1.7079  .86  295 

39 

22 

.50553  .58591  1.7067  .86281 

38 

23 

.50578  .58631  1.7056  .86266 

37 

24 

.50603  .58670  1.7045  .86251 

36 

25 

.50628  .58709  1.7033  .86237 

35 

26 

.50654  .58748  1.7022  .86222 

34 

27 

.50679  .58787  1.7011  .86207 

33 

28 

.50704  .58826  1.6999  .86192 

32 

29 

.50729  .58865  1.6988  .86178 

31 

3O 

.50  754  .58  905  1.6977  .86  163 

30 

31 

.50779  .58944  1.6965  .86148 

29 

32 

.50804  .58983  1.6954  .86133 

28 

33 

.50829  .59022  1.6943  .86119 

27 

34 

.50854  .59061  1.6932  .86104 

26 

35 

.50879  .59101  1.6920  .86089 

25 

36 

.50904  .59140  1.6909  .86074 

24 

37 

.50929  .59179  1.6898  .86059 

23 

38 

.50954  .59218  1.6887  .86045 

22 

39 

.50979  .59258  1.6875  .86030 

21 

4O 

.51004  .59297  1.6864  .86015 

2O 

41 

.51029  .59336  1.6853  .86000 

19 

42 

.51054  .59376  1.6S42  .85985 

18 

43 

.51079  .59415  1.6831  .85970 

17 

44 

.51104  .59454  1.6820  .85956 

16 

45 

.51  129  .59494  1.6808  .85941 

15 

46 

.51  154  .59533  1.6797  .85926 

14 

47 

.51  179  .59573  1.67S6  .85911 

13 

48 

.51204  .59612  1.6775  .85896 

12 

49 

.51229  .59651  1.6764  .85881 

11 

50 

.51254  .59691  1.6753  .85866 

10 

51 

.51279  .59730  1.6742  .85851 

9 

52 

.51304  .59770  1.6731  .85836 

8 

53 

.51329  .59809  1.6720  .85821 

7 

54 

.51354  .59849  1.6709  .85806 

6 

55 

.51379  .59888  1.6698  .85792 

5 

56 

.51404  .59928  1.6687  .85777 

4 

57 

.51429  .59967  1.6676  .85762 

3 

58 

.51454  .60007  1.6665  .85747 

2 

59 

.51479  .60046  1.6654  .85732 

1 

60 

.51504  .60086  1.6643  .85717 

O 

/ 

cos    cot    tan    sin 

/ 

59° 

31° 

/ 

sin    tan    cot    cos 

/ 

0 

.51504  .60086  1.6643  .85717 

60 

1 

.51  529  .60  126  1.6632  .85  702 

59 

2 

.51  554  .60  165  1.6621  .85  687 

58 

3 

.51579  .60205  1.6610  .85672 

57 

4 

•51604  .60245  1.6599  .85657 

56 

5 

.51  628  .60  284  1.6588  .85  642 

55 

6 

.51653  .60324  1.6577  .85627 

54 

'7 

.51678  .60364  1.6566  .85612 

53 

8 

.51703  .60403  1.6555  .85597 

52 

9 

.51728  .60443  1.6545  .85582 

51 

1O 

.51753  .60483  1.6534  .85567 

50 

11 

.51  778  .60522  1.6523  .85551 

49 

12 

.51803  .60562  1.6512  .85536 

48 

13 

.51828  .60602  1.6501  .85521 

47 

14 

.51852  .60642  1.6490  .85506 

46 

15 

.51877  .60681  1.6479  .85491 

45 

16 

.51902  .60721  1.6469  .85476 

44 

17 

.51927  .60761  1.6458  .85461 

43 

18 

.51952  .60801  1.6447  .85446 

42 

19 

.51977  .60841  1.6436  .85431 

41 

2O 

.52  002  .60  881  1.6426  .85  416 

40 

21 

.52026  .60921  1.6415  .85401 

39 

22 

.52051  .60960  1.6404  .85385 

38 

23 

.52076  .61000  1.6393  .85370 

37 

24 

.52  101  .61  040  1.6383  .85  355 

36 

25 

.52  126  .61  080  1.6372  .85  340 

35 

26 

.52151  .61  120  1.6361  .85325 

34 

27 

.52175  .61160  1.6351  .85310 

33 

28 

.52  200  .61  200  1.6340  .85  294 

32 

29 

.52  225  .61  240  1.6329  .85  279 

31 

30 

.52  250  .61  280  1.6319  .85  264 

30 

31 

.52  275  .61  320  1.6308  .85  249 

29 

32 

.52  299  .61  360  1.6297  .85  234 

28 

33 

.52  324  .61  400  1.6287  .85  218 

27 

31 

.52  349  .61  440  1.6276  .85  203 

26 

35 

.52374  .61480  1.6265  .85188 

25 

36 

.52  399  .61  520  1.6255  .85  173 

24 

37 

.52  423  .61  561  1.6244  .85  157 

23 

38 

.52  448  .61  601  1.6234  .85  142 

22 

39 

.52473  .61641  1.6223  .85  127 

21 

4O 

.52498  .61681  1.6212  .85112 

2O 

41 

.52  522  .61  721  1.6202  .85  096 

19 

42 

.57  547  .61  761  1.6191  .85  081 

18 

43 

.52572  .61801  1.6181  .85066 

17 

44 

.52  597  .61  842  1.6170  .85  051 

16 

45 

.52  621  .61  882  1.6160  .85  035 

15 

46 

.52  646  .61  922  1.6149  .85  020 

14 

47 

.52671  .61962  1.6139  .85005 

13 

48 

.52696  .62003  1.6128  .84989 

12 

49 

.52720  .62043  1.6118  .84974 

11 

5O 

.52745  .62083  1.6107  .84959 

1O 

51 

.52770  .62  124  1.6097  .84943 

9 

52 

.52  794  .62  164  1.6087  .84  928 

8 

53 

.52  819  .62  204  1.6076  .84  913 

7 

54 

.52  844  .62  245  1.6066  .84  897 

6 

55 

.52  869  .62  285  1.6055  .84  882 

5 

56 

.52893  .62325  1.6045  .84866 

4 

57 

.52918  .62366  1.6034  .84851 

3 

58 

.52  943  .62  406  1.6024  .84  836 

2 

59 

.52967  .62446  1.6014  .84820 

1 

60 

.52992  .62487  1.6003  .84805 

0 

/ 

cos    cot   tan    sin 

/ 

58° 

74 


NATURAL   FUNCTIONS 


32° 

/ 

sin        tan        cot        cos 

/ 

~0 

.52992  .62487  1.6003  .84805 

6O 

1 

.53017  .62527  1.5993  .84789 

59 

2 

.53041  .62568  1.5983  .84774 

58 

3 

.53  066  .62  608  1.5972  .84  759 

57 

4 

.53  091  .62  649  1.5962  .84  743 

56 

5 

.53  115  .62  689  1.5952  .84  728 

55 

6 

.53  140  .62  730  1.5941  .84  712 

54 

7 

.53164  .62770  1.5931  .84697 

53 

8 

.53189  .62811  1.5921  .84681 

52 

9 

.53214  .62852  1.5911  .84666 

51 

10 

.53  238  .62  892  1.5900  .84  650 

50 

11 

.53  263  .62  933  1.5890  .84  635 

49 

12 

.53288  .62973  1.5880  .84619 

48 

13 

.53312  .63014  1.5869  .84604 

47 

14 

.53337  .63055  1.5859  .84588 

46 

15 

.53  361  .63  095  1.5849  .84  573 

45 

16 

.53  386  .63  136  1.5839  .84  557 

44 

17 

.53411  .63177  1.5829  .84542 

43 

18 

.53435  .63217  1.5818  .84526 

42 

19 

•53460  .63258  1.5808  .84511 

41 

2O 

.53  484  .63  299  1.5798  .84  495 

40 

21 

.53  509  .63  340  1.5788  .84  480 

39 

22 

.53534  .63380  1.5778  .84464 

38 

23 

.53558  .63421  1.5768  .84448 

37 

24 

.53583  .63462  1.5757  .84433 

36 

25 

.53607  .63503  1.5747  .84417 

35 

26 

.53632  .63544  1.5737  .84402 

34 

27 

.53  656  .63  584  1.5727  .84  386 

33 

28 

.53681  .63625  1.5717  .84370 

32 

29 

.53705  .63666  1.5707  .84355 

31 

3D 

.53  730  .63  707  1.5697  .84  339 

30 

31 

.53  754  .63  748  1.5687  .84  324 

29 

32 

.53  779  .63  789  1.5677  .84  308 

28 

33 

.53  804  .63  830  1.5667  .84  292 

27 

34 

.53  828  .63  871  1.5657  .84  277 

26 

35 

.53853  .63912  1.5647  .84261 

25 

36 

.53  877  .63  953  L5637  .84  245 

24 

37 

.53902  .63994  1.5627  .84230 

23 

38 

.53926  .64035  1.5617  .84214 

22 

39 

.53951  .64076  1.5607  .84198 

21 

4O 

.53975  .64117  1.5597  .84182 

20 

41 

.54  000  .64  158  1.5587  .84  167 

19 

42 

.54024  .64199  1.5577  .84151 

18 

43 

.54  049  .64  240  1.5567  .84  135 

17 

44 

.54073  .64281  1.5557  .84120 

16 

45 

.54097  .64322  1.5547  .84104 

15 

46 

.54122  .64363  1.5537  .84088 

14 

47 

.54146  .64404  1.5527  .84072 

13 

48 

.54171  .64446  1.5517  .84057 

12 

49 

.54  195  .64  487  1.5507  .84  041 

11 

50 

.54220  .64528  1.5497  .84025 

10 

51 

.54244  .64569  1.5487  .84009 

9 

52 

.54269  .64610  1.5477  .83994 

8 

53 

.54293  .64652  1.5468  .83978 

7 

54 

.54317  .64693  1.5458  .83962 

6 

55 

.54342  .64734  1.5448  .83946 

5 

56 

.54  366  .64  775  1.5438  .83  930 

4 

57 

.54  391  .64  817  1.5428  .83  915 

3 

58 

.54415  .64858  1.5418  .83899 

2 

59 

.54  440  .64  899  1.5408  .83  883 

1 

6O 

.54  464  .64  941  1.5399  .83  867 

O 

/ 

cos        cot        tan        sin 

/ 

57° 

33° 

/ 

sin   tan   cot   cos 

/ 

0 

.54464  .64941  1.5399  .83867 

6O 

1 

.54488  .64982  1.5389  .83851 

59 

2 

.54  513  .65  024  1.5379  .83  835 

58 

3 

.54  537  .65  065  1.5369  .83  819 

57 

4 

.54  561  .65  106  1.5359  .83  804 

56 

5 

.54  586  .65  148  1.5350  .83  788 

55 

6 

.54610  .65  189  1.5340  .83772 

54 

7 

.54  635  .65  231  1.5330  .83  756 

53 

8 

.54  659  .65  272  1.5320  .83  740 

52 

9 

.54683  .65314  1.5311  .83724 

51 

10 

.54  708  .65  355  1.5301  .83  708 

50 

11 

.54  732  .65  397  1.5291  .83  692 

49 

12 

.54  756  .65  438  1.5282  .83  676 

48 

13 

.54781  .65480  1.5272  .83660 

47 

14 

.54805  .65521  1.5262  .83645 

46 

15 

.54  829  .65  563  1.5253  .83  629 

45 

16 

.54  854  .65  604  1.5243  .83  613 

44 

17 

.54  878  .65  646  1.5233  .83  597 

43 

18 

.54  902  .65  688  1.5224  .83  581 

42 

19 

.54  927  .65  729  1.5214  .83  565 

41 

2O 

.54951  .65771  1.5204  .83549 

40 

21 

.54975  .65813  1.5195  .83533 

39 

22 

.54999  .65854  1.5185  .83517 

38 

23 

.55  024  .65  896  1.5175  .83  501 

37 

24 

.55  048  .65  938  1.5166  .83  485 

36 

25 

.55072  .65980  1.5156  .83469 

35 

26 

.55097  .66021  1.5147  .83453 

34 

27 

.55121  .66063  1.5137  .83437 

33 

28 

.55-145  .66105  1.5127  .83421 

32 

29 

.55  169  .66  147  1.5118  .83  405 

31 

30 

.55  194  .66  189  1.5108  .83  389 

30 

31 

.55  218  .66  230  1.5099  .83  373 

29 

32 

.55  242  .66  272  1.5089  .83  356 

28 

33 

.55266  .66314  1.5080  .83340 

27 

34 

.55291  .66356  1.5070  .83324 

26 

35 

.55315  .66398  1.5061  .83308 

25 

36 

.55339  .66440  1.5051  .83292 

24 

37 

.55  363  .66  482  1.5042  .83  276 

23 

38 

.55  388  .66  524  1.5032  .83  260 

22 

39 

.55412  .66566  1.5023  .83244 

21 

40 

.55  436  .66  608  1.5013  .83  228 

20 

41 

.55460  .66650  1.5004  .83212 

19 

42 

.55484  .66692  1.4994  .83195 

18 

43 

.55  509  .66  734  1.4985  .83  179 

17 

44 

.55  533  .66  776  1.4975  .83  163 

16 

45 

.55557  .66818  1.4966  .83147 

15 

46 

.55581  .66860  1.4957  .83131 

14 

47 

.55605  .66902  1.4947  .83115 

13 

48 

.55630  .66944  1.4938  .83098 

12 

49 

.55654  .66986  1.4928  .83082 

11 

50 

.55  678  .67  028  1.4919  .83  066 

10 

51 

.55  702  .67  071  1.4910  .83  050 

9 

52 

.55726  .67113  1.4900  .83034 

8 

53 

.55750  .67155  1.4891  .83017 

7 

54 

.55  775  .67  197  1.4882  .83  001 

6 

55 

.55  799  .67  239  1.4872  .82  985 

5 

56 

.55823  .67282  1.4863  .82969 

4 

57 

.55847  .67324  1.4854  .82953 

3 

58 

.55871  .67366  1.4844  .82936 

2 

59 

.55  895  .67  409  1.4835  .82  920 

1 

60 

.55919  .67451  1.4826  .82904 

0 

/ 

cos    cot    tan    sin 

/ 

56° 

NATURAL   FUNCTIONS 


75 


34° 

/ 

sin        tan        cot        cos 

/ 

o 

.55919  .67451  1.4826  .82904 

6O 

1 

.55  943  .67  493  1.4816  .82  887 

59 

2 

.55968  .67536  1.4807  .82871 

58 

3 

.55992  .67578  1.4798  .82855 

57 

4 

.56016  .67620  1.4788  .82839 

56 

5 

.56  040  .67  663  1.4779  .82  822 

55 

6 

.56064  .67705  1.4770  .82806 

54 

7 

.56088  .67748  1.4761  .82790 

53 

8 

.56112  .67790  1.4751  .82773 

52 

9 

.56  136  .67  832  1.4742  .82  757 

51 

10 

.56  160  .67  875  1.4733  .82  741 

50 

11 

.56  184  .67  917  1.4724  .82  724 

49 

12 

.56  208  .67  960  1.4715  .82  708 

48 

13 

.56232  .68002  1.4705  .82692 

47 

14 

.56  256  .68  045  1.4696  .82  675 

46 

15 

.56280  .68088  1.4687  .82659 

45 

16 

.56  305  .68  130  1.4678  .82  643 

44 

17 

.56  329  .68  173  1.4669  .82  626 

43 

18 

.56353  .68215  1.4659  .82610 

42 

19 

.56377  .68258  1.4650  .82593 

41 

2O 

.56401  .68301  1.4641  .82577 

40 

21 

.56425  .68343  1.4632  .82561 

39 

22 

.56  449  .68  386  1.4623  .82  544 

38 

23 

.56473  .68429  1.4614  .82528 

37 

24 

.56497  .68471  1.4605  .82511 

36 

25 

.56521  .68514  1.4596  .82495 

35 

26 

.56545  .68557  1.4586  .82478 

34 

27 

.56569  .68600  1.4577  .82462 

33 

28 

.56593  .68642  1.4568  .82446 

32 

29 

.56617  .68685  1.4559  .82429 

31 

30 

.56641  .68728  1.4550  .82413 

3O 

31 

.56665  .68771  1.4541  .82396 

29 

32 

.56689  .68814  1.4532  .82380 

28 

33 

.56713  .68857  1.4523  .82363 

27 

34 

.56736  .68900  1.4514  .82347 

26 

35 

.56760  .68942  1.4505  .82330 

25 

36 

.56784  .68985  1.4496  .82314 

24 

37 

.56808  .69028  1.4487  .82297 

23 

38 

.56832  .69071  1.4478  .82281 

22 

39 

.56856  .69114  1.4469  .82264 

21 

4O 

.56880  .69157  1.4460  .82248 

2O 

41 

.56904  .69200  1.4451  .82231 

19 

42 

.56928  .69243  1.4442  .82214 

18 

43 

.56952  .69286  1.4433  .82198 

17 

44 

.56  976  .69  329  1.4424  .82  181 

16 

45 

.57  000  .69  372  1.4415  .82  165 

15 

46 

.57024  .69416  1.4406  .82148 

14 

47 

.57047  .69459  1.4397  .82132 

13 

48 

.57071  .69502  1.4388  .82115 

12 

49 

.57095  .69545  1.4379  .82098 

11 

50 

.57119  .69588  1.4370  .82082 

1O 

51 

.57143  .69631  1.4361  .82065 

9 

52 

.57167  .69675  1.4352  .82048 

8 

53 

.57191  .69718  1.4344  .82032 

7 

54 

.57215  .69761  1.4335  .82015 

6 

55 

.57238  .69804  1.4326  .81999 

5 

56 

.57262  .69847  1.4317  .81982 

4 

57 

.57286  .69891  1.4308  .81965 

3 

58 

.57310  .69934  1.4299  .81949 

2 

59 

.57334  .69977  1.4290  .81932 

1 

6O 

.57358  .70021  1.4281  .81915 

O 

/ 

cos        cot        tan        sin 

/ 

55° 

35° 

/ 

sin        tan        cot        cos 

/ 

O 

.57358  .70021  1.4281  .81915 

60 

1 

.57381  .70064  1.4273  .81899 

59 

2 

.57405  .70107  1.4264  .81882 

58 

3 

.57429  .70151  1.4255  .81865 

57 

4 

.57  453  .70  194  1.4246  .81  848 

56 

5 

.57  477  .70  238  1.4237  .81  832 

55 

6 

.57  501  .70  281  1.4229  .81  815 

54 

7 

.57  524  .70  325  1.4220  .81  798 

53 

8 

.57548  .70368  1.4211  .81782 

52 

9 

.57572  .70412  1.4202  .81765 

51 

10 

.57596  .70455  1.4193  .81748 

50 

11 

.57619  .70499  1.4185  .81731 

49 

12 

.57643  .70542  1.4176  .81714 

48 

13 

.57667  .70586  1.4167  .81698 

47 

14 

.57691  .70629  1.4158  .81681 

46 

15 

.57715  .70673  1.4150  .81664 

45 

16 

.57738  .70717  1.4141  .81647 

44 

17 

.57  762  .70  760  1.4132  .81  631 

43 

18 

.57786  .70804  1.4124  .81614 

42 

19 

.57810  .70848  1.4115  .81597 

41 

20 

.57833  .70891  1.4106  .81580 

40 

21 

.57857  .70935  1.4097  .81563 

39 

22 

.57881  .70979  1.4089  .81546 

38 

23 

.57904  .71023  1.4080  .81530 

37 

24 

.57  928  .71  066  1.4071  .81  513 

36 

25 

.57952  .71110  1.4063  .81496 

35 

26 

.57976  .71154  1.4054  .81479 

34 

27 

.57999  .71198  1.4045  .81462 

33 

28 

.58  023  .71  242  1.4037  .81  445 

32 

29 

.58  047  .71  285  1.4028  .81  428 

31 

3O 

.58070  .71329  1.4019  .81412 

30 

31 

.58  094  .71  373  1.4011  .81  395 

29 

32 

.58  118  .71  417  1.4002  .81  378 

28 

33 

.58  141  .71  461  1.3994  .81  361 

27 

34 

.58  165  .71  505  1.3985  .81  344 

26 

35 

.58189  .71549  1.3976  .81327 

25 

36 

.58212  .71593  1.3968  .81310 

24 

37 

.58236  .71637  1.3959  .81293 

23 

38 

.58260  .71681  1.3951  .81  276 

22 

39 

.58  283  .71  725  1.3942  .81  259 

21 

4O 

.58307  .71769  1.3934  .81242 

20 

41 

.58330  .71813  1.3925  .81225 

19 

42 

.58354  .71857  1.3916  .81208 

18 

43 

.58378  .71901  1.3908  .81191 

17 

44 

.58401  .71946  1.3899  .81174 

16 

45 

.58425  .71990  1.3891  .81157 

15 

46 

.58  449  .72  034  1.3882  .81  140 

14 

47 

.58472  .72078  1.3874  .81  123 

13 

48 

.58  496  .72  122  1.3865  .81  106 

12 

49 

.58  519  .72  167  1.3857  .81  089 

11 

5O 

.58543  .72211  1.3848  .81072 

1O 

51 

.58567  .72255  1.3840  .81055 

9 

52 

.58590  .72299  1.3831  .81038 

8 

53 

.58614  .72344  1.3823  .81021 

7 

54 

.58637  .72388  1.3814  .81004 

6 

55 

.58661  .72432  1.3806  .80987 

5 

56 

.58684  .72477  1.3798  .80970 

4 

57 

.58708  .72521  1.3789  .80953 

3 

58 

.58731  .72565  1.3781  .80936 

2 

59 

.58755  .72610  1.3772  .80919 

1 

60 

.58  779  .72  654  1.3764  .80  902 

0 

/ 

cos        cot        tan        sin 

/ 

54° 

76 


NATURAL   FUNCTIONS 


36° 

/ 

sin        tan        cot        cos 

/ 

~o 

.58779  .72654  1.3764  .80902 

6O 

i 

.58802  .72699  1.3755  .80885 

59 

2 

.58  826  .72  743  1.3747  .80  867 

58 

3 

.58  849  .72  788  1.3739  .80  850 

57 

4 

.58  873  .72  832  1.3730  .80  833 

56 

5 

.58896  .72877  1.3722  .80816 

55 

6 

.58920  .72921  1.3713  .80799 

54 

7 

.58943  .72966  1.3705  .80782 

53 

8 

.58967  .73010  1.3697  .80765 

52 

9 

.58  990  .73  055  1.3688  .80  748 

51 

10 

.59014  .73100  1.3680  .80730 

50 

11 

.59037  .73144  1.3672  .80713 

49 

12 

.59  061  .73  189  1.3663  .80  696 

48 

13 

.59084  .73234  1.3655  .80679 

47 

14 

.59108  .73.278  1.3647  .80662 

46 

IS 

.59131  .73323  1.3638  .80644 

45 

16 

.59154  .73368  1.3630  .80627 

44 

17 

.59178  .73413  1.3622  .80610 

43 

18 

.59201  .73457  1.3613  .80593 

42 

19 

.59225  .73502  1.3605  .80576 

41 

2O 

.59248  .73'  547  1.3597  .80558 

40 

21 

.59  272  .73  592  1.3588  .80  541 

39 

22 

.59  295  .73  637  1.3580  .80  524 

38 

23 

.59318  .73681  1.3572  .80507 

37 

24 

.59  342  .73  726  1.3564  .80  489 

36 

25 

.59365  .73771  1.3555  .80472 

35 

26 

.59389  .73816  1.3547  .80455 

34 

27 

.59412  .73861  1.3539  .80438 

•33 

28 

.59436  .73906  1.3531  .80420 

32 

29 

.59459  .73951  1.3522  .80403 

31 

3O 

.59482  .73996  1.3514  .80386 

30 

31 

.59506  .74041  1.3506  .80368 

29 

32 

.59529  .74086  1.3498  .80351 

28 

33 

.59552  .74131  1.3490  .80334 

27 

34 

.59576  .74176  1.3481  .80316 

26 

35 

.59599  .74221  1.3473  .80299 

25 

36 

.59622  .74267  1.3465  .80282 

24 

37 

.59646  .74312  1.3457  .80264 

23 

38 

.59669  .74357  1.3449  .80247 

22 

39 

.59693  .74402  1.3440  .80230 

21 

40 

.59716  .74447  1.3432  .80212 

2O 

41 

.59739  .74492  1.3424  .80195 

19 

42 

.59763  .74538  1.3416  .80178 

18 

43 

.59  786  .74  583  1.3408  .80  160 

17 

44 

.59809  .74628  1.3400  .80143 

16 

45 

.59832  .74674  1.3392  .80125 

15 

46 

.59  856  .74  719  1.3384  .80  108 

14 

47 

.59  879  .74  764  1.3375  .80  091 

13 

48 

.59902  .74810  1.3367  .80073 

12 

49 

.59926  .74855  1.3359  .80056 

11 

50 

.59949  .74900  1.3351  .80038 

1O 

51 

.59972  .749-16  1.3343  .80021 

9 

52 

.59995  .74991  1.3335  .80003 

8 

53 

.60019  .75037  1.3327  .79986 

7 

54 

.60042  .75082  1.3319  .79968 

6 

55 

.60065  .75128  1.3311  .79951 

5 

56 

.60089  .75173  1.3303  .79934 

4 

57 

.60112  .75219  1.3295  .79916 

3 

58 

.60135  .75264  1.3287  .79899 

2 

59 

.60158  .75310  1.3278  .79881 

1 

60 

.60  182  .75  355  1.3270  .79  864 

O 

/ 

cos        cot        tan        sin 

/ 

53° 

37° 

/ 

sin   tan   cot   cos 

/ 

O 

.60182  .75355  1.3270  .79864 

60 

1 

.60205  .75401  1.3262  .79846 

59 

2 

.60228  .75447  1.3254  .79829 

58 

3 

.60251  .75492  1.3246  .79811 

57 

4 

.60  274  .75  538  1.3238  .79  793 

.56 

5 

.60  298  .75  584  1.3230  .79  776 

55 

6 

.60321  .75629  1.3222  .79758 

54 

7 

.60344  .75675  1.3214  .79741 

53 

8 

.60  367  .75  721  1.3206  .79  723 

52 

9 

.60390  .75767  1.3198  .79706 

51 

1O 

.60414  .75812  1.3190  .79688 

50 

11 

.60437  .75858  1.3182  .79671 

49 

12 

.60460  .75904  1.3175  .79653 

48 

13 

.60483  .75950  1.3167  .79635 

47 

14 

.60506  .75996  1.3159  .79618 

46 

15 

.60529  .76042  1.3151  .79600 

45 

16 

.60553  .76088  1.3143  .79583 

44 

17 

.60576  .76134  1.3135  .79565 

43 

18 

.60599  .76180  1.3127  .79547 

42 

19 

.60622  .76226  1.3119  .79530 

41 

2O 

.60645  .76272  1.3111  .79512 

4O 

21 

.60668  .76318  1.3103  .79494 

39 

22 

.60691  .76364  1.3095  .79477 

38 

23 

.60714  .76410  1.3087  .79459 

37 

24 

.60738  .76456  1.3079  .79441 

36 

25 

.60761  .76502  1.3072  .79424 

35 

26 

.60784  .76548  1.3064  .79406 

34 

27 

.60807  .76594  1.3056  .79388 

33 

28 

.60830  .76640  1.3048  .79371 

32 

29 

.60853  .76686  1.3040  .79353 

31 

3O 

.60876  .76733  1.3032  .79335 

30 

31 

.60899  .76779  1.3024  .79318 

29 

32 

.60922  .76825  1.3017  .79300 

28 

33 

.60945  .76871  1.3009  .79282 

27 

34 

.60968  .76918  1.3001  .79264 

26 

35 

.60991  .76964  1.2993  .79247 

25 

36 

.61015  .77010  1.2985  .79229 

24 

37 

.61038  .77057  1.2977  .79211 

23 

38 

.61  061  .77  103  1.2970  .7?  193 

22 

39 

.61  084  .77  149  1.2962  .79  176 

21 

4O 

.61  107  .77  196  1.2954  .79  158 

20 

41 

.61  130  .77242  1.2946  .79140 

19 

42 

.61  153  .77  289  1.2938  .79  122 

18 

43 

.61  176  .77335  .2931  .79105 

17 

44 

.61199  .77382  .2923  .79087 

16 

45 

.61222  .77428  .2915  .79069 

15 

46 

.61245  .77475  .2907  .79051 

14 

47 

.61  268  .77  521  .2900  .79  033 

13 

•48 

.61  291  .77  568  .2892  .79  016 

12 

49 

.61314  .77615  1.28S4  .78998 

11 

50 

.61337  .77661  1.2876  .78980 

1O 

51 

.61  360  .77  708  1.2869  .Z8_%4 

9 

52 

.61383  .77754  1.2861  ./*94^ 

8. 

53 

.61406  .77801  1.2853  .78926 

7 

54 

.61  429  .77  848  1.2846  .78  90S 

6 

55 

.61451  .77895  1.2838  .78891 

5 

56 

.61  474  .77  941  1.2830  .78  873 

4 

57 

.61497  .77988  1.2822  .78855 

3 

58 

.61520  .78035  1.2815  .78837 

2 

59 

.61543  .78082  1.2807  .78819 

1 

00 

.61  566  .78  129  1.2799  .78  801 

O 

/ 

cos    cot    tan    sin 

/ 

52° 

NATURAL   FUNCTIONS 


38° 

f 

sin   tan   cot   cos 

/ 

o 

.61566  .78129  1.2799  .78801 

60 

1 

.61  589  .78  175  1.2792  .78  783 

59 

2 

.61  612  .78  222  1.2784  .78  765 

58 

3 

.61  635  .78  269  1.2776  .78  747 

57 

4 

.61658  .78316  1.2769  .78729 

56 

5 

.61681  .78363  1.2761  .78711 

55 

6 

.61  704  .78  410  1.2753  .78  694 

54 

7 

.61  726  .78  457  1.2746  .78  676 

53 

8 

.61749  .78504  1.2738  .78658 

52 

9 

..61772  .78551  1.2731  .78640 

51 

1O 

.61  795  .78  598  1.2723  .78  622 

50 

11 

.61818  .78645  1.2715  .78604 

49 

12 

.61  841  .78  692  1.2708  .78  586 

48 

13 

.61  864  .78  739  1.2700  .78  568 

47 

14 

.61  887  .78  786  1.2693  .78  550 

46 

15 

.61  909  .78  834  1.2685  .78  532 

45 

16 

.61932  .78881  1.2677  .78514 

44 

17 

.61  955  .78  928  1.2670  .78  496 

43 

18 

.61  978  .78  975  1.2662  .78  478 

42 

19 

.62001  .79022  1.2655  .78460 

41 

2O 

.62  024  .79  070  1.2647  .78  442 

4O 

21 

.62046  .79117  1.2640  .78424 

39 

22 

.62069  .79164  1.2632  .78405 

38 

23 

.62092  .79212  1.2624  .78387 

37 

24 

.62115  .79259  1.2617  .78369 

36 

25 

.62138  .79306  1.2609  .78351 

35 

26 

.62  160  .79  354  1.2602  .78  333 

34 

27 

.62183  .79401  1.2594  .78315 

33 

28 

.62  206  .79  449  1.2587  .78  297 

32 

29 

.62229  .79496  1.2579  .78279 

31 

30 

.62251  .79544  1.2572  .78261 

30 

31 

.62  274  .79  591  1.2564  .78  243 

29 

32 

.62297  .79639  1.2557  .78225 

28 

33 

.62  320  .79  686  1.2549  .78  206 

27 

34 

.62  342  .79  734  1.2542  .78  188 

26 

35 

.62  365  .79  781  1.2534  .78  170 

25 

36 

.62388  .79829  1.2527  .78152 

24 

37 

.62411  .79877  1.2519  .78134 

23 

38 

.62433  .79924  1.2512  .78116 

22 

39 

.62456  .79972  1.2504  .78098 

21 

40 

.62479  .80020  1.2497  .78079 

2O 

41 

.62502  .80067  1.2489  .78061 

19 

42 

.62524  .80115  1.2482  .78043 

18 

43 

.62547  .80163  1.2475  .78025 

17 

44 

.62570  .80211  1.2467  .78007 

16 

45 

.62592  .80258  1.2460  .77988 

15 

46 

.62615  .80306  1.2452  .77970 

14 

47 

.62638  .80354  1.2445  .77952 

13 

48 

.62660  .80402  1.2437  .77934 

12 

49 

.62683  .80450  1.2430  .77916 

11 

50 

.62  706  .80  498  1.2423  .77  897 

1O 

51 

.62  728  .80  546  1.2415  .77  879 

9 

52 

.62  751  .80  594  1.2408  .77  861 

8 

53 

.62  774  .80  642  1.2401  .77  843 

7 

54 

.  .62  796  .80  690  1.2393  .77  824 

6 

55 

.62819  .80738  1.23S6  .77806 

5 

56 

.62  842  .80  786  1.2378  .77  788 

4 

57 

.62864  .80834  1.2371  .77769 

3 

58 

.62887  .80882  1.2364  .77751 

2 

59 

.62909  .80930  1.2356  .77733 

1 

6O 

.62932  .80978  1.2349  .77715 

O 

/ 

cos    cot   tan    sin 

/ 

51° 

39° 

/ 

sin    tan    cot    cos 

/ 

O 

.62932  .80978  1.2349  .77715 

60 

1 

.62955  .81027  1.2342  .77696 

59 

2 

.62977  .81075  1.2334  .77678 

58 

3 

.63000  .81123  1.2327  .77660 

57 

4 

.63  022  .81  171  1.2320  .77  641 

56 

5 

.63045  .81220  1.2312  .77623 

55 

6 

.63  068  .81  268  1.2305  .77  605 

54 

7 

.63  090  .81  316  1.2298  .77  586 

53 

8 

.63  113  .81  364  1.2290  .77  568 

52 

9 

.63  135  .81  413  1.2283  .77  550 

51 

10 

.63  158  .81  461  1.2276  .77  531 

50 

11 

.63180  .81510  1.2268  .77513 

49 

12 

.63  203  .81  558  1.2261  .77  494 

48 

13 

.63  225  .81  606  1.2254  .77  476 

47 

14 

.63248  .81655  1.2247  .77458 

46 

15 

.63  271  .81  703  1.2239  .77  439 

45 

16 

.63  293  .81  752  1.2232  .77  421 

44 

17 

.63  316  .81  800  1.2225  .77  402 

43 

18 

.63  338  .81  849  1.2218  .77  384 

42 

19 

.63  361  .81  898  1.2210  .77  366 

41 

20 

.63  383  .81  946  1.2203  .77  347 

40 

21 

.63  406  .81  995  1.2196  .77  329 

39 

22 

.63  428  .82  044  1.2189  .77  310 

38 

23 

.63451  .82092  1.2181  .77292 

37 

24 

.63473  .82141  1.2174  .77273 

36 

25 

.63  496  .82  190  1.2167  .77  255 

35 

26 

.63  518  .82  238  1.2160  .77  236 

34 

27 

.63  540  .82  287  1.2153  .77  218 

33 

28 

.63  563  .82  336  1.2145  .77  199 

32 

29 

.63  585  .82  385  1.2138  .77  181 

31 

30 

.63608  .82434  1.2131  .77162 

3O 

31 

.63  630  .82  483  1.2124  .77  144 

29 

32 

.63653  .82531  1.2117  .77125 

28 

33 

.63  675  .82  580  1.2109  .77  107 

27 

34 

.63698  .82629  1.2102  .77088 

26 

35 

.63  720  .82  678  1.2095'  .77  070 

25 

36 

.63742  .82727  1.2088  .77051 

24 

37 

.63765  .82776  1.2081  .77033 

23 

38 

.63  787  .82  825  1.2074  .77  014 

22 

39 

.63  810  .82  874  1.2066  .76  996 

21 

4O 

.63832  .82923  1.2059  .76977 

2O 

41 

.63854  .82972  1.2052  .76959 

19 

42 

.63877  .83022  1.2045  .76940 

18 

43 

.63  899  .83  071  1.2038  .76  921 

17 

44 

.63  922  .83  120  1.2031  .76  903 

16 

45 

.63944  .83169  1.2024  .76884 

15 

46 

.63966  .83218  1.2017  .76866 

14 

47 

.63989  .83268  1.2009  .76847 

13 

48 

.64011  .83317  1.2002  .76828 

12 

49 

.64  033  .83  366  1.1995  .76  810 

11 

50 

.64056  .83415  1.1988  .76791 

10 

51 

.64078  .83465  1.1981  .76772 

9 

52 

.64  100  .83  514  1.1974  .76  754 

8 

53 

.64  123  .83  564  1.1967  .76  735 

7 

54 

.64145  .83613  1.1960  .76717 

6 

55 

.64  167  .83  662  1.1953  .76  698 

5 

56 

.64  190  .83  712  1.1946  .76  679 

4 

57 

.64212  .83761  1.1939  .76661 

3 

58 

.64234  .83811  1.1932  .76642 

2 

59 

.64256  .83860  1.1925  .76623 

1 

6O 

.64279  .83910  1.1918  .76604 

O 

/ 

cos    cot   tan    sin 

/ 

50° 

78 


NATURAL   FUNCTIONS 


40° 

/ 

sin    tan    cot   cos 

/ 

0 

.64279  .83910  1.1918  .76604 

60 

1 

.64301  .83960  1.1910  .76586 

59 

2 

.64323  .84009  1.1903  .76567 

58 

3 

.64346  .84059  1.1896  .76548 

57 

4 

.64368  .84108  1.1889  .76530 

56 

5 

.64390  .84158  1.18S2  .76511 

55 

6 

.64412  .84208  1.1875  .76492 

54 

7 

.64435  .84258  1.1868  .76473 

53 

8 

.64457  .84307  1.1861  .76455 

52 

9 

.64479  .84357  1.1854  .76436 

51 

1O 

.64501  .84407  1.1847  .76417 

50 

11 

.64524  .84457  1.1840  .76398 

49 

12 

.64546  .84507  1.1833  .76380 

48 

13 

.64568  .84556  1.1826  .76361 

47 

14 

.64590  .84606  1.1819  .76342 

46 

15 

.64612  .84656  1.1812  .76323 

-45 

16 

.64  635  .84  706  1.1806  .76  304 

44 

17 

.64  657  .84  756  1.1799  .76  286 

43 

18 

.64679  .84806  1.1792  .76267 

42 

19 

.64701  .84856  1.1785  .76248 

41 

2O 

.64723  .81-906  1.1778  .76229 

4O 

21 

.64746  .84956  .1771  .76210 

39 

22 

.64  768  .85  006  1.1764  .76  192 

38 

23 

.64790  .85057  .1757  .76173 

37 

24 

.64  812  .85  107  .1750  .76  154 

36 

25 

.64834  .85157  .1743  .76135 

35 

26 

.64856  .85207  .1736  .76116 

34 

27 

.64878  .85257  1.1729  .76097 

33 

28 

.64901  .85308  1.1722  .76078 

32 

29 

.64923  .85358  1.1715  .76059 

31 

3O 

.64945  .85408  1.1708  .76041 

30 

31 

.64967  .85458  1.1702  .76022 

29 

32 

.64  989  .85  509  1.1695  .76  003 

28 

33 

.65011  .85559  1.1688  .75984 

27 

34 

.65  033  .85  609  1.1681  .75  965 

26 

35 

.65055  .85660  1.1674  .75946 

25 

36 

.65  077  .85  710  1.1667  .75  927 

24 

37 

.65  100  .85  761  1.1660  .75  908 

23 

38 

.65  122  .85  811  1.1653  .75  889 

22 

39 

.65  144  .85  862  1.1647  .75  870 

21 

4O 

.65  166  .85912  1.1640  .75851 

20 

41 

.65  188  .85  963  1.1633  .75  832 

19 

42 

.65  210  .86  014  1.1626  .75  813 

18 

43 

.65  232  .86064  1.1619  .75  794 

17 

44 

.65254  .86115  1.1612  .75775 

16 

45 

.65  276  .86  166  1.1606  .75  756 

15 

46 

.65  298  .86  216  1.1599  .75  738 

14 

47 

.65  320  -86  267  1.1592  .75  719 

13 

48 

.65342  .86318  1.1585  .75700 

12 

49 

.65364  .86368  1.1578  .75680 

11 

5O 

.65386  .86419  1.1571  .75661 

10 

51 

.65408  .86470  1.1565  .75642 

9 

52 

•65430  .86521  1.1558  .75623 

8 

53 

.65452  .86572  1.1551  .75604 

7 

54 

.65474  .86623  1.1544  .75585 

,  6 

55 

.65496  .86674  1.1538  .75566 

5 

56 

.65518  .86725  1.1531  .75547 

4 

57 

.65  540  .86  776  1.1524  .75  528 

3 

58 

.65562  .86827  1.1517  .75509 

2 

59 

.65584  .86878  1.1510  .75490 

1 

6O 

.65606  .86929  1.1504  .75471 

o 

/ 

cos    cot    tan    sin 

/ 

49° 

41° 

/ 

sin    tan    cot    cos 

/ 

0 

.65606  .86929  1.1504  .75471 

60 

1 

.65  628  .86  980  1.1497  .75  452 

59 

2 

.65650  .87031  1.1490  .75433 

58 

3 

.65  672  .87  082  1.1483  .75  414 

57 

4 

.65  694  .87  133  1.1477  .75  395 

56 

5 

.65  716  .87  184  1.1470  .75  375 

55 

6 

.65  738  .87  236  1.1463  .75  356 

54 

7 

.65  759  .87  287  1.1456  .75  337 

53 

8 

.65781  .87338  1.1450  .75318 

52 

9 

.65803  .87389  1.1443  .75299 

51 

1O 

.65  825  .87  441  1.1436  .75  280 

50 

11 

.65  847  .87  492  1.1430  .75  261 

49 

12 

.65  869  .87  543  1.1423  .75  241 

48 

13 

.65  891  .87  595  1.1416  .75  222 

47. 

14 

.65913  .87646  1.1410  .75203 

46 

15 

.65  935  .87  698  1.1403  .75  184 

45 

16 

.65956  .87  749  1.1396  .75  165 

44 

17 

.65  978  .87  801  1.1389  .75  146 

43 

18 

.66000  .87852  1.1383  .75  126 

42 

19 

.66022  .87904  1.1376  .75107 

41 

2O 

.66044  .87955  1.1369  .75088 

4O 

21 

.66066  .88007  1.1363  .75069 

39 

22 

.66088  .88059  1.1356  .75050 

38 

23 

.66109  .88110  1.1349  .75030 

37 

24 

.66131  .88162  1.1343  .75  QU^ 

36 

25 

.66153  .88214  1.1336  .74992 

35 

26 

.66175  .88265  1.1329  .74973 

34 

27 

.66197  .88317  1.1323  .74953 

•33 

28 

.66218  .88369  1.1316  .74934 

32 

29 

.66240  .88421  1.1310  .74915 

31 

30 

.66262  .88473  1.1303  .74896 

30 

31 

.66  284  .88  524  1.1296  .74  876 

29 

32 

.66306  .88576  1.1290  .74857 

28 

33 

.66327  .88628  1.1283  .74838 

27 

34 

.66349  .88680  1.1276  .74818 

26 

35 

.66371  .88732  1.1270  .74799 

25 

36 

.66393  .88784  1.1263  .74780 

24 

37 

.66414  .88836  1.1257  .74760 

23 

38 

.66436  .88888  1.1250  .74741 

22 

39 

.66458  .88940  1.1243  .74722 

21 

4O 

.66480  .88992  1.1237  .74703 

2O 

41 

.66501  .89045  1.1230  .74683 

19 

42 

.66523  .89097  1.1224  .74664 

18 

43 

.66545  .89149  1.1217  .74644 

17 

44 

.66566  .89201  1.1211  .74625 

16 

45 

.66588  .89253  1.1204  .74606' 

15 

46 

.66610  .89306  1.1197  .74586 

14 

47 

.66632  .89358  1.1191  .74567 

13 

48 

.66653  .89410  1.1184  .74548 

12 

49 

.66675  .89463  1.1178  .74528 

11 

50 

.66697  .89515  1.1171  .74509 

1O 

51 

.66718  .89567  1.1165  .74489 

9 

52 

.66  740  .89  620  1.1158  .74  470 

8 

53 

.66762  .89672  1.1152  .74451 

7 

54 

.66783  .89725  1.1145  .74431 

6 

55 

.66805  .89777  1.1139  .74412 

5 

56 

.66827  .89830  1.1132  .74392 

4 

57 

.66848  .89883  1.1126  .74373 

3 

58 

.66870  .89935  1.1119  .74353 

2 

59 

.66891  .89988  1.1113  .74334 

1 

6O 

.66913  .90040  1.1106  .74314 

O 

/ 

cos    cot   tan    sin 

/ 

48° 

NATURAL  FUNCTIONS 


79 


42° 

/ 

sin    tan    cot    cos 

/ 

o 

.66913  .90040  1.1106  .74314 

60 

1 

.66935  .90093  1.1100  .74295 

59 

2 

.66956  .90146  1.1093  .74276 

58 

3 

.66  978  .90  199  1.1087  .74  256 

57 

4 

.66999  .90251  1.1080  .74237 

56 

5 

.67021  .90304  1.1074  .74217 

55 

6 

.67043  .90357  1.1067  .74198 

54 

7 

.67064  .90410  1.1061  .74178 

53 

8 

.67086  .90463  1.1054  .74159 

52 

9 

.67107  .90516  1.1048  .74139 

51 

1O 

.67  129  .90  569  1.1041  .74  120 

50 

11 

.67151  .90621  1.1035  .74100 

49 

12 

.67172  .90674  1.1028  .74080 

48 

13 

.67194  .90727  1.1022  .74061 

47 

14 

.67  215  .90  781  1.1016  .74  041 

46 

15 

.67  237  .90  834  1.1009  .74  022 

45 

16 

.67258  .90887  1.1003  .74002 

44 

17 

.67  280  .90  940  1.0996  .73  983 

43 

18 

.67301  .90993  1.0990  .73963 

42 

19 

.67323  .91046  1.0983  .73944 

'  41 

20 

.67344  .91099  1.0977  .73924 

4O 

21 

.67366  .91153  1.0971  .73904 

39 

22 

.67  387  .91  206  1.0964  .73  885 

38 

23 

.67409  .91259  1.0958  .73865 

37 

24 

.67430  .91313  1.0951  .73846 

36 

25 

.67  452  .91  366  1.0945  .73  826 

35 

26 

.67473  .91419  1.0939  .73806 

34 

27 

.67495  .91473  1.0932  .73787 

33 

28 

.67516  .91526  1.0926  .73767 

32 

29 

.67  538  .91  580  1.0919  .73  747 

31 

3O 

.67  559  .91  633  1.0913  .73  728 

30 

31 

.67  580  .91  687  1.0907  .73  708 

29 

32 

.67  602  .91  740  1.0900  .73  688 

28 

33 

.67  623  .91  794  1.0894  .73  669 

27 

34 

.67  645  '.91  847  1.0888  .73  649 

26 

35 

.67  666  .91  901  1.0881  .73  629 

25 

36 

.67  688  .91  955  1.0875  .73  610 

24 

37 

.67709  .92008  1.0869  .73590 

23 

38 

.67  730  .92  062  1.0862  .73  570 

22 

39 

.67  752  .92  116  1.0856  -.73  551 

21 

40 

.67  773  .92  170  1.0850  .73  531 

2O 

41 

.67795  .92224  1.0843  .73511 

19 

42 

.67  816  .92  277  1.0837  .73  491 

18 

43 

.67837  .92331  1.0831  .73472 

17 

44 

.67859  .92385  1.0824  .73452 

,16 

45 

.67880  .92439  1.0818  .73432 

15 

46 

.67901  .92493  1.0812  .73413 

14 

47 

.67923  .92547  1.0805  .73393 

13 

48 

.67  944  .92  601  1.0799  .73  373 

12 

49 

.67965  .92655  1.0793  .73353 

11 

50 

.67  987  .92  709  1.0786  .73  333 

1O 

51 

.68008  .92763  1.0780  .73314 

9 

52 

.68029  .92817  1.0774  .73294 

8 

53 

.68051  .92872  1.0768  .73274 

7 

54 

.68  072  .92  926  1.0761  .73  254 

6 

55 

.68093  .92980  1.0755  .73234 

5 

56 

.68  115  .93  034  1.0749  .73  215 

4 

57 

.68  136  .93  088  1.0742  .73  195 

3 

58 

.68  157  .93  143  1.0736  .73  175 

2 

59 

.68  179  .93  197  1.0730  .73  155 

1 

60^ 

.68  200  .93  252  1.0724  .73  135 

O 

/ 

cos   cot   tan   sin 

/ 

47° 

43° 

/ 

sin        tan        cot        cos 

/ 

0 

.68200  .93252  1.0724  .73135 

6O 

1 

.68221  .93306  1.0717  .73116 

59 

2 

.68242  .93360  1.0711  .73096 

58 

3 

.68  264  .93  415  1.0705  .73  076 

57 

4 

.68285  .93469  1.0699  .73056 

56 

5 

.68  306  .93  524  1.0692  .73  036 

55 

6 

.68327  .93578  1.0686  .73016 

54 

7 

.68  349  .93  633  1  .0680  .72  996 

53 

8 

.68  370  .93  688  1.0674  .72  976 

52 

9 

.68  391  .93  742  1.0668  .72  957 

51 

1O 

.68412  .93797  1.0661  .72937 

50 

11 

.68434  .93852  1.0655  .72917 

49 

12 

.68455  .93906  1.0649  .72897 

48 

13 

.68476  .93961  1.0643  .72877 

47 

14 

.68497  .94016  1.0637  .72857 

46 

15 

.68518  .94071  1.0630  .72837 

45 

16 

.68  539  .94  125  1.0624  .72  817 

44 

17 

.68  561  .94  180  1.0618  .72  797 

43 

18 

.68582  .94235  1.0612  .72777 

42 

19 

.68603  .94290  1.0606  .72757 

41 

2.0 

.68624  .94345  1.0599  .72737 

40 

21 

.68645  .94400  1.0593  .72717 

39 

22 

.68666  .94455  1.0587  .72697 

38 

23 

.68688  .94510  1.0581  .72677 

37 

24 

.68  709  .94  565  1.0575  .72  657 

36 

25     .68  730  .94  620  1.0569  .72  637 

35 

26    .68751  .94676  1.0562  .72617 

34 

27     .68  772  .94  731  1.0556  .72  597 

33 

28 

.68  793  .94  786  1.0550  .72  577 

32 

29 

.68814  .94841  1.0544  .72557 

31 

30 

.68835  .94896  1.0538  .72537 

30 

31 

.68  857  .94  952.  1.0532  .72  517 

29 

32 

.68878  .95007  1.0526  .72497 

28 

33 

.63_S.9_9  .95062  1.0519  .72477 

27 

34 

.68920  .95118  1.0513  .72457 

26 

35 

.68941  .95173  1.0507  .72437 

25 

36 

.68962  .95229  1.0501  .72417 

24 

37 

.68  983  .95  284  1.0495  .72  397 

23 

38 

.69004  .95340  1.0489  .72377 

22 

39 

.69025  .95395  1.0483  .72357 

21 

40 

.69046  .95451  1.0477  .72337 

2O 

41 

.69067  .95506  1.0470  .72317 

19 

42 

.69  088  .95  562  1.0464  .72  297 

18 

43 

.69  109  .95  618  1.0458  .72  277 

17 

44 

.69  130  .95  673  1.0452  .72  257 

16 

45 

.69  151  .95  729  1.0446  .72  236 

15 

46 

.69172  .95785  1.0440  .72216 

14 

47 

.69  193  .95  841  1.0434  .72  196 

13 

48 

.69  214  .95  897  1.0428  .72  176 

12 

49 

.69235  .95952  1.0422  .72156 

11 

50 

.69256  .96008  1.0416  .72136 

1O 

51 

.69  277  .96  064  1.0410  .72  116 

9 

52 

.69  298  .96  120  1.0404  .72  095 

8 

53 

.69319  .96176  1.0398  .72075 

7 

54 

.69340  .96232  1.0392  .72.055 

6 

55 

.69361  .96288  1.0385  .72035 

5 

56 

.69382  .96344  1.0379  .72015 

4 

57 

.69403  .96400  1.0373  .71995 

3 

58 

.69424  .96*457  1.0367  .71974 

2 

59 

.69445  .96513  1.0361  .71954 

1 

6O 

.69466  .96569  1.0355  .71934 

O 

/ 

cos        cot        tan        sin 

/ 

46° 

80 


NATURAL  FUNCTIONS 


44° 

/ 

sin   tan   cot   cos 

/ 

o 

.69  466  .96  569  1.0355  .71  934 

6O 

1 

.69487  .96625  1.0349  .71914 

59 

2 

.69508  .96681  1.0343  .71894 

58 

3 

.69  529  .96  738  .1.0337  .71  873 

57 

4 

.69549  .96794  1.0331  .71853 

56 

5 

.69  570  .96  850  1.0325  .71  833" 

55 

6 

.69591  .96907  1.0319  .71813 

54 

7 

.69612  .96963  1.0313  .71792 

53 

8 

.69633  .97020  1.0307  .71772 

52 

9 

.69654  .97076  1.0301  .71752 

51 

10 

.69  675  .97  133  1.0295  .71  732 

50 

11 

.69696  .97189  1.0289  .71711 

49 

12 

.69717  .97246  1.0283  .71691 

48 

13 

.69737  .97302  1.0277  .71671 

47 

14 

.69758  .97359  1.0271  .71650 

46 

15 

.69  779  .97  416  1.0265  .71  630 

45 

16 

.69800  .97472  1.0259  .71610 

44 

17 

.69821  .97529  1.0253  .71590 

43 

18 

.69842  .97586  1.0247  .71569 

42 

19 

.69862  .97643  1.0241  .71549 

41 

2O 

.69883  .97700  1.0235  .71529 

4O 

21 

.69904  .97756  1.0230  .71508 

39 

22 

.69925  .97813  1.0224  .71488 

38 

23 

.69946  .97870  1.0218  .71468 

37 

24 

.69966  .97927  1.0212  .71447 

36 

25 

.69987  .97984  1.0206  .71427 

35 

26 

.70008  .98041  1.0200  .71407 

34 

27 

.70029  .98098  1.0194  .71386 

33 

28 

.70049  .98155  1.01SS  .71366 

32 

29 

.70070  .98213  1.0182  .71345 

31 

30 

.70091  .98270  1.0176  .71325 

30 

31 

.70112  .98327  1.0170  .71305 

29 

32 

.70  132  .98  384  1.0164  .71  284 

28 

33 

.70153  .98441  1.0158  .71264 

27 

34 

.70174  .98499  1.0152  .71243 

26 

35 

.70195  .98556  1.0147  .71223 

25 

36 

.70215  .98613  1.0141  .71203 

24 

37 

.70  236  .98  671  1.0135  .71  182 

23 

38 

.70257  .98728  1.0129  .71162 

22 

39 

.70277  .98786  1.0123  .71141 

21 

40 

.70298  .98843  1.0117  .71  121 

2O 

41 

.70319  .98901  1.0111  .71100 

19 

42 

.70339  .98958  1.0105  .71080 

18 

43 

.70360  .99016  1.0099  .71059 

17 

44 

.70381  .99073  1.0094  .71039 

16 

45 

.70401  .99131  1.0088  .71019 

15 

46 

.70422  .99189  1.0082  .70998 

14 

47 

.70443  .99247  1.0076  .70978 

13 

48 

.70463  .99304  1.0070  .70957 

12 

49 

.70484  .99362  1.0064  .70937 

11 

5O 

.70505  .99420  1.0058  .70916 

1O 

51 

.70525  .99478  1.0052  .70896 

9 

52 

.70546  .99536  1.0047  .70875 

8 

53 

.70567  .99594  1.0041  .70855 

7 

54 

.70587  .99652  1.0035  .70834 

6 

55 

.70608  .99710  1.0029  .70813 

5 

56 

.70628  .99768  1.0023  .70793 

4 

57 

.70649  .99826  1.0017  .70772 

3 

58 

.70670  .99884  1.0012  .70752 

2 

59 

.70690  .99942  1.0006  .70731 

1 

6O 

.70711  1.0000  1.0000  .70711 

O 

/ 

cos    cot   tan    sin 

.  / 

45° 

n          I 


7 


• 


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